7007.tp.indd 1 6/24/08 10:25:24 AMAxioms for Lattices and Boolean Algebras This page intentionally left blankThis page intentionally left blank N E W… [621828]
7007.tp.indd 1 6/24/08 10:25:24 AMAxioms for Lattices and
Boolean Algebras
This page intentionally left blankThis page intentionally left blank
N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I World ScientificR Padmanabhan
University of Manitoba, Canada
S Rudeanu
University of Bucharest, Romania
7007.tp.indd 2 6/24/08 10:25:26 AMAxioms for Lattices and
Boolean Algebras
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN-13 978-981-283-454-6
ISBN-10 981-283-454-0All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd.Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Printed in Singapore.AXIOMS FOR LATTICES AND BOOLEAN ALGEBRAS
LaiFun – Axioms for Lattices.pmd 6/17/2008, 2:46 PM 1
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Introduction
Ever since Euclid the axiomatic approach is at the heart of mathematics.
The axiomatic approach admits the possibility of a mixture of deductive and
empirical reasoning and hence it is an ideal pedagogical tool. Also, in the
emerging 21st century it is the natural choice of modern theorem-provers
for development and experimentation of automated reasoning. Among the
various types of axioms one can formulate for a given theory, identities are
the most natural ones. Many familiar algebraic systems occurring in lattice
theory are usually de ned by means of equational identities, i.e., sentences
in the form f=g, wherefandgare formed from variables and symbols
denoting the fundamental operations of the relevant algebras. The purpose
of this monograph is to collect and present all known minimal equational
bases for semilattices, lattices, modular lattices and Boolean algebras.
There is a huge literature on the axioms of several equational classes of
lattices from 1880 onwards. The original 1963 monograph by Rudeanu {
the genesis of this monograph { reports the state of the art of the subject
at that time. The present book updates the original monograph in several
respects. We report not only new axiom systems { and there are a lot! {
but also several deep metatheorems (i.e., theorems about axiom systems)
that have been proved in the meantime. Unlike the old monograph, the
present one includes many proofs. Besides, the strategy in presenting old
papers has been changed. Let us explain all this in some detail.
The rst four chapters of the book present systems of axioms for semi-
lattices and lattices, modular lattices, distributive lattices, and Boolean
algebras and orthomodular lattices, no matter whether they are given in
terms of join and meet or in terms of other tools, such as a ternary oper-
ation, a ternary relation, or others. In the case of Boolean algebras most
systems use complementation (either as a basic operation or in a disguised
form) together with join and meet or with join only, while other systems
de ne Boolean algebras in terms of a single binary operation (the \She er
v
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
vi Introduction
stroke") or in terms of ring operations; we have included all of them, as
well as the axioms of the related concept of a Boolean group. A few related
lines of research are sketched in Chapter 5, where we also suggest several
open problems.
In order to keep the dimension of our book within reasonable limits, we
have in general not included systems that characterize a class of lattices
within a larger class. In other words, we have almost exclusively collected
systems that do not reproduce all the axioms of a class of lattices larger
than the one being de ned. The most notable exceptions are a few charac-
terizations of modular lattices (Chapter 2, x1) and of distributive lattices
(Chapter 3, x1), which are immediately obtained from the de nitions, and
a major exception, the characterization of Boolean algebras within the class
of all uniquely complemented lattices (Chapter 4, x8). Here we address the
celebrated problem of E.V. Huntingron (1904), which, according to a lead-
ing expert in modern lattice theory, is one of the two problems that shaped
a century of research in lattice theory (cf. George Gr atzer). The problem
was whether every uniquely complemented lattice is distributive. Hunting-
ton believed that every such lattice was distributive, and hence Boolean.
He himself gave some sucient conditions which force a uniquely comple-
mented lattice to be Boolean. Birkho and von Neumann proved that
modularity is one such property, i.e., every uniquely complemented modu-
lar lattice is distributive. Although Huntington's conjecture was disproved
by Dilworth in 1945 (he established that every lattice is isomorphic with
a sublattice of a lattice with unique complements), the interest for nding
conditions which ensure distributivity of uniquely complemented lattices
has remained intact. In Chapter 4 we show that there are uncountably
many non-modular lattice identities which force a uniquely complemented
lattice to be Boolean, thus providing several new axiom systems for Boolean
algebras within the class of all uniquely complemented lattices.
We have included, to the best of our knowledge, all of the papers that
suit the above description. We have actually described in the book the
most signi cant systems in our appreciation; this has resulted, for instance,
in more than 40 systems for distributive lattices or bounded distributive
lattices and 70 systems for Boolean algebras. However, certain authors
provided hundreds or thousands of axioms systems (!), as shown in the
book. Some of the systems we have chosen are given with proofs and some
of the proofs in the book are new, without mention of this fact.
Beside the ve chapters, our monograph comprises six appendices.
Appendix A, written by W. McCune, reproduces four proofs provided by
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Introduction vii
a computer program called Prover9. Appendix B is a bibliography on
axiom systems for partially ordered systems and betweenness in posets.
Appendix C is a very short presentation of quasilattices, a class of alge-
bras (A; _;^) that captures the essence of all regular identities in lattices.
Appendix D compiles a bibliography of papers devoted to the axiomatics
of Lukasiewicz-Moisil algebras, which play an important role in algebraic
logic. Appendix E lists a few papers which suggest methods for testing
the associativity of a binary operation. Several papers that deal with E.H.
Moore's complete existential theory (i.e., a kind of exhaustive analysis of
all existing implications that link a set of conditions) are brie
y presented
in Appendix F.
The Bibliography refers to the four chapters, while each appendix has
its own bibliography.
While the 1963 monograph also aimed at exhaustiveness in the sense
described above, the present monograph has several new features. One of
them was already mentioned: the inclusion of many proofs.
A manifest tendency in the literature is the search for minimal equa-
tional bases (i.e., systems of equational identities with the smallest possible
number of axioms) for equational classes of lattices and in particular the
search for de nitions by a single equational identity whenever such single
identities exist. A new feature in this monograph is the application of some
structural properties in discovering minimal equational bases. Thus, for
example, while the de nability of groups by a single axiom depends upon
the type in which groups are de ned (cf. Tarski-Green, 1968), all nitely-
based varieties of orthomodular lattices (in particular, Boolean algebras)
are always one-based, whatever the type, thanks to the permutable and
distributive congruence properties.
Another new feature is to exploit the self-dual nature of the classes of
lattices dealt with in order to give independent self-dual sets of axioms.
Last but not least, another new feature of this monograph is the em-
phasis on the computer program Otter and its improved version Prover9,
which turn out to be very ecient theorem-provers.
At this point certain technical explanations seem necessary.
The papers referred to in this book cover a period of more than hundred
years. As the mathematical style has meanwhile altered to a certain extent,
the realization of a unitary presentation in our book raised certain problems.
Thus in very old papers axioms were not understood just as properties of
the operations, but they also included the very existence of these operations;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
viii Introduction
for instance, any system of axioms for lattices began with something like
\With any x;y2Ais associated an element x_y2A" and a similar
axiom for ^. Nowadays the role of such \existence axioms", as they were
called, is played by the genus proximus, which is an algebra of a certain
signature. In our book we refer to these early papers as if they had been
written according to contemporary standards, like \A lattice is an algebra
(A;_;^) of type (2,2) such that :::" followed by the remaining axioms of
the author. This strategy is exactly the opposite of the one adopted in the
monograph by Rudeanu [1963].
A curiosity of certain old papers is that they include pairs of axioms of
the formpandp=)q, instead of simply listing axioms pandq. This was
probably due to the desire of facilitating the proof of the independence of
axiomp: any model in which axiom pfails automatically satis es axiom
p=)q. We have taken the liberty of ignoring these arti ces, by replacing
axiomp=)qby axiomq. To be sure, in all these circumstances we say
nothing about the independence of the system, even if the original system
was independent.
For each type of statement, the displayed statements of this book are
numbered separately according to the usual Statement m:n:p convention.
For instance, Proposition m:n:p means the p{th proposition in Section n
of Chapter m. However within Chapter mthe statements m:n:p may be
referred to simply as n:p.
Formulae are numbered in each section by a single number; formula
(n:p) means formula (p) in Section nof the current chapter, while (m:n:p)
is formula (n:p) in Chapter m.
The notation Author [year]* indicates papers that are not available in
our libraries and which we quote from other sources, mainly from Mathe-
matical Reviews.
Acknowledgements. We sincerely thank George Gr atzer for the keen
interest he took in this project ever since it was conceived. Needless to
say that R. Padmanabhan bene ted a lot by periodically presenting his
discoveries in Dr. Gr atzer's seminar over the past 40 years. We also
thank George for inserting a pre-publication announcement of this book
in his forthcoming survey on lattice theory \Two problems that shaped
a century of lattice theory". We thank Dr. David Kelly for reading the
Introduction and making some constructive comments which enhanced our
presentation. R. Padmanabhan also thanks all his collaborators (David
Kelly, Harry Lakser, William McCune, Craig Platt, R.W. Quackenbush,
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Introduction ix
Robert Vero and Barry Wolk) for having stimulating discussions on lattice
axioms over the years and to Dr. Guenter Krause, Head of the Department
of Mathematics, University of Manitoba for creating a pleasant atmosphere
and camaraderie conducive for doing productive research. We thank Bill
McCune for sending us a tex-formatted Prover9 proofs which form Appen-
dix A. S. Rudeanu wishes to thank Dan A. Simovici, who during several
years urged him to write this book. We also thank Laurent iu Leu stean,
who provided us with quite a lot of papers.
The authors deeply appreciate the editorial sta at World Scienti c for
their meticulous attention to details and we express our sincere gratitude to
Ms. Lai Fun Kwong for her expert advice and for the careful proof-reading.
R. Padmanabhan's research was supported by an ongoing operation
grant from NSERC of Canada for the past 38 years.
Last but certainly not least, we are extremely grateful to Cristian S.
Calude for his kind appreciation of our book, for his e orts in promoting
it and for his invaluable \TeXnical" help, without which, our monograph
would have never made it this far. Saying \Thank you, Cris" would be a
massive understatement.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
This page intentionally left blankThis page intentionally left blank
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Contents
Introduction v
1. Semilattices and Lattices 1
2. Modular Lattices 39
3. Distributive Lattices 53
4. Boolean Algebras 69
5. Further Topics and Open Problems 137
Appendix A: Some Prover9 Proofs 147
Appendix B: Partially Ordered Sets and Betweenness 159
Appendix C: Quasilattices 181
Appendix D: Lukasiewicz-Moisil Algebras 183
Appendix E: Testing Associativity 187
Appendix F: Complete Existential Theory and Related Concepts 189
Bibliography 193
Index 211
xi
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
This page intentionally left blankThis page intentionally left blank
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Chapter 1
Semilattices and Lattices
Alattice is a structure (L; ), where is a partial order on L, that is, it
satis es the properties
P1 8x(xx);
P2 8x8y(xy&yx=)x=y);
P3 8x8y8z(xy&yz=)xz);
(re
exivity, antisymmetry andtransitivity), and every two elements have a
least upper bound and a greatest lower bound:
P4 8x8y9s(xs&ys& (xz&yz=)sz));
P5 8x8y9p(px&py& (zx&zy=)zp)):
It is easily shown that the elements sandpare uniquely determined by x
andy, which enables one to de ne
(D) x_y=s; x^y=p:
The system of axioms fP1;P2;P3;P4;P5gis due to Peirce [1880-84] and
Ore [1935], who corrected certain
aws in the original system of Peirce.
Sorkin [1951] proved the independence of the system. Another system is due
to Bennett [1930], who used axioms P 1;P2;P3and two more complicated
variants of P 4and P 5.
More generally, a structure (L; ) is called a join semilattice if it satis es
P1;P2;P3;P4, and is known as a meet semilattice if it satis es P 1;P2;P3;P5.
Since the elements sandpin (D) are uniquely determined by xandy,
this opens the way to the de nition of semilattices and lattices as algebras,
whose axioms will be studied in x1 and xx2,3, respectively. The concept
of identity, which is crucial in this book, is carefully explained in x1 in the
particular case of groupoids. In x4 we present systems of axioms for lat-
tices in terms of lattice betweenness, segments or partially de ned ternary
operations.
1
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2 1. Semilattices and Lattices
1.1. Semilattices
It is easily seen that the operation _of a join semilattice and the oper-
ation^of a meet semilattice are idempotent, commutative andassociative.
This remark has led to the concept of semilattice as a groupoid. Recall
that a groupoid or an algebra of type (2) (S; ) is a setSendowed with a
binary operation :SS !S. So a semilattice is a groupoid (S; )
which satis es the identities
S1xx=x;
S2xy=yx;
S3 (xy)z=x(yz):
This concept is due to Klein-Barmen [1934]. See also the historical notice
in Birkho [1948], Ch.II, footnote 6, but note that the name Halbverband
had been used earlier, in Klein-Barmen [1939].
The independence of the system of axioms fS1;S2;S3gwas established
e.g. in Sorkin [1951], Dubreil-Jacotin, Lesieur and Croisot [1953] and
Rudeanu [1959].
It is immediately seen that the following identities hold in a semilattice:
S4 (uv)((wx)(yz)) = ((u v)(xw))(zy);
S5 ((xy)z)t=t(x(yz));
S6 (xy)z= (yz)x;
S7x(yz) =z(xy):
They have been used in several two-axiom characterizations of semilattices,
namely fS1;S4g;fS1;S5g;fS1;S6gandfS1;S7g, due to Potts [1965], Petcu
[1967], Padmanabhan [1966] and Soboci nski [1979], respectively.
Let us prove that the last two systems actually de ne semilattices. Since
S6and S 7reduce to S 3in presence of commutativity, it suces to prove S 2.
It follows from S 1and S 6that
yz= (yz)(yz) = (z (yz))y= ((yz)y)z
= ((zy)y)z= ((yy)z)z= (yz)z= (zz)y=zy;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.1 Semilattices 3
while S 1and S 7imply
yz=y(zz) =z(yz) =z((yy)z) =z(z(yy))
=z(y(zy)) = (z y)(zy) =zy:2
Ruedin [1966], [1966/67] remarked that the system fS1;S2;S3gcan be
replaced by fS1;S2;S8;S9g, where
S8x(yx) =yx;
S9x(yz) = (x y)(xz) ;
the proof is a re nement of the well-known proof that transforms (S; )
into a join/meet semilattice. From this Ruedin derived a characterization
of semilattices with neutral element, previously obtained by Felscher and
Klein-Barmen [1959].
There is one more equivalent de nition of semilattices. To state it we
need a few preliminaries.
Let (S; ) be a groupoid and let Vbe a set of elements called variables .
The setV[f; (;)gis called alphabet and its elements are said to be letters ;
aword is a concatenation of letters. The set of expressions or terms
is the least set of words obeying the following rules: 1) every variable is a
term, and 2) if 'and are terms, then (') ( ) is a term; however
we writexinstead of (x) for every variable xoccurring in a compound
expression. In universal algebraic terminology, the terms are simply the
elements of the clone<>generated by . Thus every term 'is obtained
in nitely many steps by applying the above rules 1) and 2). The variables
x1;:::;x noccurring in this construction of 'are known as the variables of
'; we also say that 'is an expression in the variables x1;:::;x n.
The crucial point is that each term'generates a function with argu-
ments and values in S, which is obtained by interpreting each letter x2V
occurring in 'as a variable in the usual sense, and each occurrence of the
letter as the symbol of the binary operation of the algebra (S; ). The
functions generated by terms are called term functions orpolynomials.
An identity is currently meant as something like
8×1:::8xnf(x1;:::; n) =g(x1;:::;x n) ;
however in axiomatics we refer to identities that may or may not be ful-
lled. The exact meaning of this alternative is the following. Whenever
the concept of expression is naturally de ned over an algebra Alike in the
above particular case of groupoids, by an identity'= is meant in fact a
notation for a pair ('; ) of terms'; ; we say that A satis es this identity
if the terms '; generate the same polynomial.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4 1. Semilattices and Lattices
Now we are going to prove the promised result.
Lemma 1.1.1. In a semilattice (S;)every term in n variables
x1;:::;x ngenerates the function x1 xn.
Proof: By algebraic induction on the expression '. If'is a variable
x1, the property is trivial. Suppose '= , where the terms and
satisfy the property. Let y1;:::;y pandz1;:::;z qbe the variables of
and , respectively. Then fy1;:::;y pg [ fz 1;:::;z qg=fx1;:::;x ngand
= (y1 yp)(z1 zq) =x1:::x nby S1;S2and S 3. 2
Lemma 1.1.2. If the groupoid (S;)satis es S1, the polynomial f generated
by a term satis es f(x;:::;x) = x.
Proof: Again by algebraic induction. The inductive step follows from
( )(x;:::;x) = (x;:::;x) (x;:::;x) = xx=x:
2
Theorem 1.1.1. (Petcu [1971]). A groupoid (S;)is a semilattice if and
only if every two expressions in the same variables generate the same
function.
We are going to prove a slight re nement of this theorem, which needs
the following introduction. A polynomial in nvariables is said to be es-
sentially n-ary if it depends actually upon all of its nvariables. Let Pn(S)
denote the number of essentially n-ary polynomials of a groupoid S.
Theorem 1.1.10The following conditions are equivalent for a groupoid
(S;):
(i)Pn(S) = 1 for all n;
(ii)Pn(S) = 1 forn:= 1; 2;3;
(iii) (S; ) is a semilattice.
Proof: (i)=)(ii): Trivial.
(ii)=)(iii): Applying P1(S) = 1 we get xx=xbecause both xand
xxare unary. Similarly, by applying P2(S) = 1 we obtain commutativity
sincexyandyxmust coincide, and nally P3(S) = 1 forces associativity.
(iii)=)(i): By Lemma 1.1. 2
Recently G. Gr atzer asked whether semilattices can be de ned by sys-
tems containing identities of arbirary length. The armative answer is
provided by the following theorem, which was proved by Padmanabhan
and Wolk in the early 1970's, but has so far remained unpublished.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.1 Semilattices 5
Theorem 1.1.2. For anyn>2; ;S1and
(SLn)x1(x2(x3 (xn 2(xn 1xn)):::)
=x2(x3 (xn 1(xnx1)):::);
form an independent system of axioms de ning semilattices.
Proof: The system fS1;SL3g de nes semilattices because by applying
SL3 twice we obtain S 7:x(yz) =y(zx) =z(xy). The rest of the
proof consists in showing that SLn =)SL(n 1). To simplify notation we
will use concatenation instead of and the shortcut x3(x4(:::(xn 3(xn 2=
Y. So SLn reads
(1) x1(x2(Y(xn 1xn):::) =x2(Y(xn 1(xnx1):::):
By takingxn:=xn 1:=x1and using idempotency S 1we obtain
(2) x1(x2(Yx1):::) =x2(Yx1):::);
and by taking further xn 2:=xn 3:=:=x3:=x1we get
(3) x1(x2x1) =x2x1:
On the other hand, taking x1:=xn 1xnin (1) yields
(4) (xn 1xn) (x2(Y(xn 1xn):::) =x2(Y(xn 1(xn(xn 1xn)):::):
Now takex1:=xn 1xnin (2) and use in turn (4), (3), (1) and S 1; then
x2(Y(xn 1xn):::) = (x n 1xn) (x2(Y(xn 1xn):::)
=x2(Y(xn 1(xn(xn 1xn):::) =x2(Y(xn 1(xn 1xn):::)
=xn(x2(Y(xn 1xn 1):::) =xn(x2(Y x n 1):::);
whence we obtain SL(n 1) forxn:=x1.
To prove the independence of SLn de nexy=x, while with xy=
constant one obtains the independence of S 1. 2
A similar question was answered by Tarski [1968], who proved that
semilattices can be de ned by independent systems of as many axioms as
desired. His proof is purely existential and is based on topological methods.
We give below a constructive proof.
First we introduce a notation. Given a groupoid (S; ) we de ne
(5) <1>x=x; <n + 1>x=x<n>x; n 2Nnf0g:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
6 1. Semilattices and Lattices
Lemma 1.1.3. Supposea;b2Nnf0g and identities < a + 1> x =
<b+ 1>x=xhold. Ifc= g:c:d:fa;bg then<c+ 1>x=x.
Proof: Recall that the Euclidean algorithm for nding ccan be given
the following \subtractive" formy: Suppose e.g. that a>b. De ne c0=a
and whilecibsetci+1=ci b. Thenc=cq, whereqis the index such
thatcq<b.
It follows that <c0+1>x=xand ifiqsatis es<ci 1+1>x=x
then
<ci+ 1>x=x<ci>x=<b + 1>x<ci 1 b>x
=<b + 1 +ci 1 b>x =<c i 1+ 1>x=x:
Therefore<ci+ 1>x=x(i= 0;1;:::;q ), implying <c+ 1>x=x.2
Theorem 1.1.3. For everyn2there is an independent system of n
identities de ning semilattices.
Proof: Forn:= 2 we have already given several systems of two axioms,
whose independence is easy to establish.
Now for every n2 we are going to construct an independent system
ofn+ 1 identities, the last one being S 7.
Letp1;p2;:::;p nbe the rst nprimes and P=p1:::p n. De neqi=
P=p i(i= 1;:::;n). We claim that
(6) f<q i+ 1>x=x(i= 1;:::;n); x (yz) =z(xy)g
is an independent system de ning semilattices.
Setr1=q1andri= g:c:d:fr i 1;qig(i= 2;:::;n). Then < r1+ 1>
x=xand if<ri 1+ 1>x=xthen Lemma 1.3 implies <ri+ 1>x=x.
Therefore<ri+ 1>x=xfori= 1;:::;n; in particular <rn+ 1>x=
x. But according to a well-known property, rn= g:c:d:fq 1;:::;q ng= 1,
therefore<2>x=x, which is S 1.
To prove independence, take rst S:=Z3=f0;1;2gandxy= 2x +2y,
where the operations are taken modulo 3. Then the commutative operation
is not associative, because (x y)z=x+y+2z, whilex(yz) = 2x +y+z.
Butxx=x, hence< m > x =xfor allm. Therefore axiom S 7is
independent.
yUseful in computer science!
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.2 De ning Lattices in Terms of the Operations _and^ 7
To proceed further, for each i2 f1;:::;ng consider the eld (Z pi;+;),
where Zpi=f0;:::;pi 1gand the operations are taken modulo pi. Then
<m>x =x++x=mx, therefore
<m>x =0()m=0()pijm:
Consequently < qi> x6=0, hence< qi+ 1> x6=x, whereas for every
j6=iwe have< qj+ 1> x =< q j> x+x=x. Therefore each axiom
<qi+ 1>x=xis independent. 2
The above theorem also provides a new answer to the question addressed
in the previous theorem: the lengths of identities in independent bases for
semilattices may be as large as one pleases. So these are \unbounded" in
every sense of the term.
1.2. De ning Lattices in Terms of the Operations _and^
It is easily seen that the operations _(join) and ^(meet) of a lattice
(cf. P 4;P5and (1)) are idempotent, commutative, associative and satisfy
theabsorption laws, i.e.,
L_
1x_x=x;
L^
1x^x=x;
L_
2x_y=y_x;
L^
2x^y=y^x;
L_
3 (x_y)_z=x_(y_z);
L^
3 (x^y)^z=x^(y^z);
L_
4x_(x^y) =x;
L^
4x^(x_y) =x:
Conversely, if an algebra (L; _;^) satis es the above properties, then it
can be proved that the equivalence
(1) x_y=y()x^y=x
holds and the relation de ned by
(2) xy()x_y=y(()x^y=x)
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
8 1. Semilattices and Lattices
satis es properties P 1 P5. Thus a lattice is equivalently de ned as an
algebra (L; _;^) satisfying the system of axioms
L0=fL_
1;L^
1;L_
2;L^
2;L_
3;L^
3;L_
4;L^
4g:
So, while in x1 we have regarded semilattices as algebras of type (2), in
this section we deal with lattices de ned as algebras of type (2,2), that is,
endowed with two binary operations _;^.
The separation of the system L0from the other properties of a Boolean
algebra was rst accomplished by Schr oder [1890-1905]. Then Dedekind
[1897] noted that axioms L_
1;L^
1can be proved from L_
4;L^
4:
x_x=x_(x^(x_x)) =x; x^x=x^(x_(x^x)) =x;
so that system L0is equivalent to
L1=fL_
2;L^
2;L_
3;L^
3;L_
4;L^
4g:
This was proved by Ore [1935]. The problem of the independence of L1was
raised by Birkho [1948] and solved in the armative by Kimura [1950].
The notation L_
i;L^
iemphasizes the pairs of dual axioms, i.e., which are
obtained from each other by interchanging _and^. We denote by L ithe
setfL_
i;L^
ig. Thus, e.g., we have just proved that L 4implies L 1.
A system of axioms is called self-dual if it consists of pairs of dual
axioms; for instance, both L0andL1are self-dual. The existence of a
self-dual system of axioms for a class of lattices implies the fact that the
principle of duality holds for that class. This means that for each theorem
valid in the lattices of that class, the dual theorem also holds: it is obtained
from the original theorem by interchanging _and^.
Sorkin [1951] inaugurated a direction of research aimed at the idea of an
exhaustive exploration of many alternatives. He considered all the variants
of the absorption laws that can be obtained by permuting the letters x;y,
namely L_
4;L^
4and
L_
5x_(y^x) =x;
L^
5x^(y_x) =x;
L_
6 (x^y)_x=x;
L^
6 (x_y)^x=x;
L_
7 (y^x)_x=x;
L^
7 (y_x)^x=x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.2 De ning Lattices in Terms of the Operations _and^ 9
and determined all the independent systems of axioms for lattice theory
that can be obtained from the larger set of axioms
1=fL_
2;L^
2;L_
3;L^
3;:::; L_
7;L^
7g;
in other words, all the independent subsystems of 1that are equivalent to
1. There are 34 such systems, each of them having 6 or 7 axioms; one of
these systems is of course the standard system L1.
Kalman [1951] generalized Sorkin's results as follows. For each of the 214
subsystems of 1he found all the subsystems of 1that are equivalent to
and all the axioms in 1that are implied by . This has applications to
the theory of skew lattices. Roughly speaking, a skew lattice is an algebra
(L;_;^) which has lattice-like properties, except the commutativity of the
two operations.
According to a well-known theorem of Birkho , a class of algebras can be
de ned by a system of identities if and only if it is closed under the formation
of subalgebras, direct products and homomophic images. Such classes of
algebras are said to be equational classes orvarieties ; for instance, the class
of all lattices is equational. The above result explains the interest of nding
equational characterizations of certain classes of algebras which have not
originally been de ned by identities, as was the case e.g. of modular lattices
and Post algebras.
Note that all of the axioms L 1;:::; L7are identities. But the above
comments do not exclude the interest of including axioms that are not
identities. Thus Sorkin [1951] introduced the following weakenings of the
absorption laws L 4and L 7:
L_
8x^y=y=)x_y=x;
L^
8x_y=y=)x^y=x;
L_
9x^y=x=)x_y=y;
L^
9x_y=x=)x^y=y;
respectively, and determined all the independent subsystems of
2=fL_
1;L^
1;L_
2;L^
2;L_
3;L^
3;L_
8;L^
8;L_
9;L^
9g
that de ne lattices. There are 18 such systems, each of them having 7
axioms; one of them is a system introduced by Klein-Barmen [1932] and
whose independence was rst proved by Kobayasi [1943].
Rudeanu [1959] solved the same problem for the larger set
3= 2[ fL_
4;L^
4g
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
10 1. Semilattices and Lattices
and refound, of course, all the systems determined by Sorkin, plus 22 new
systems, each of them having 6 or 7 axioms.
Petcu [1964] proved that lattices can be de ned using only variants of
the absorption and associative laws. Namely, he considered the axioms
L_
3;L^
3and
L_
10 (x_y)_z=x_(z_y);
L_
11 (x_y)_z=y_(x_z);
L_
12 (x_y)_z=y_(z_x);
L_
13 (x_y)_z= (y_z)_x;
L_
14 (x_y)_z= (z_y)_x;
L_
15 (x_y)_z= (x_z)_y;
L_
16x_(y_z) =y_(x_z);
L_
17x_(y_z) =z_(x_y);
L_
18x_(y_z) =z_(y_x);
and their duals L^
10;:::; L^
18. Petcu showed that these are all the distinct
variants of the associative law that can be imagined up to a permutation
of the variables x;y;z , and looked for independent subsystems of
4=fL_
4;L^
4;:::; L_
7;L^
7;L_
3;L^
3;L_
10;L^
10;:::; L_
18;L^
18g
that de ne lattices. He found 32 such systems, each of them having 4
axioms.
In the same paper Petcu introduced 80 new axioms, which he called
absorptio-associative:
((x^y)^z)_(((x^y)^z)^t) =x^(y^z)
is a sample of these axioms. He looked for independent sets of axioms for
lattices that can be constructed out of the 82-identity system 5which
consists of the idempotent laws L_
1;L^
1and the 80 absorptio-associative
axioms. He found 96 such systems, each of them having 3 axioms, namely
two absorptio-associative laws and one idempotent law. Malliah [1971]
introduced some more absorptio-associative laws and obtained 1240 new
independent sets of axioms for lattices, among which 112 have 4 identities
and 1128 consist of 3 identities. Petcu and Malliah also showed that many
other combinations do not de ne lattices, but it is not known whether
the systems they found exhaust all the minimal combinations that de ne
lattices.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.2 De ning Lattices in Terms of the Operations _and^ 11
Felscher [1957], [1958] introduced eight new variants of the associative
laws, among which
L_
19x_(y_z) = (x _y)_(x_z);
L^
19x^(y^z) = (x ^y)^(x^z);
and used them in order to construct 11 new sets of identities de ning lat-
tices. One of these sets is the self-dual system
L2=fL_
2;L^
2;L_
4;L^
4;L_
19;L^
19g:
At this point we emphasize an idea which has been used many times
in the axiomatics of lattices and Boolean algebras, as will be seen in this
book. Namely, by changing the \normal" order of letters in certain axioms,
one can prove commutativity, thus obtaining shorter systems of axioms.
So, for instance, Sz asz [1963] changed L_
19;L^
19to
L_
20x_(y_z) = (y _x)_(z_x);
L^
20x^(y^z) = (y ^x)^(z^x);
and proved that the self-dual system
L3=fL_
4;L^
4;L_
20;L^
20g
de nes lattices. He also proved the independence of L2;L3and of a system
of axioms due to Klein-Barmen [1932]. Moreover, these proofs, as well as
the proof of the independence of L1given by Dubreil-Jacotin, Lesieur and
Croisot [1953], are optimal, to the e ect that the models used in them are
of the shortest possible lengths.
Now we are going to prove that L2andL3actually de ne lattices. Since
the commutativity L 2transforms axioms L 19into L 20, it suces to do the
proof for L3.
It was recalled before that axioms L 4imply the idempotency L 1. From
L1and L 20we infer commutativity:
x_y=x_(y_y) = (y _x)_(y_x) =y_x;
and similarly for L^
2. Further we use L 4to prove
L_
21 (x_y)_x=x_y;
L^
21 (x^y)^x=x^y:
Indeed,
(x_y)_x= (x_y)_((x_y)^x) =x_y
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
12 1. Semilattices and Lattices
and similarly for L^
21. Finally we use L 20;L21and commutativity to prove
associativity, say L_
3:
(x_y)_z= (x_z)_(y_z) = (x _(y_z))_(z_(y_z))
= (x_(y_z))_(y_z) =x_(y_z):
2
It is clear that, as was remarked above, several variants of system L2
can be constructed by replacing L 4and/or L 19by variants of them such
as L5;L6;L7and L 20, respectively; variants of L 8and L 9also arise nat-
urally. Felscher (op.cit.) actually pointed out a few such sets of axioms,
but it was Ruedin [1966], [1966/67], [1967], [1967/68], [1968] who under-
took a laborious study in the same spirit as but apparently unaware of
the work by Sorkin, Kalman, Rudeanu and Petcu. It should be mentioned
that Ruedin related his research to the axiomatics of regular distributive
groupoids (as he did for semilattices; cf.x1): a groupoid (G; ) is called right
[left]distributive if it satis es the identity
(xy)z= (xz)(yz) [x(yz) = (x y)(xz) ]
andright [left]regular provided the identity
x(yx) =yx[ (xy)x=xy]
holds. Ruedin found several independent sets of axioms for lattices, namely
44/24/3 sets having 6/5/4 axioms each. He also found many independent
systems of axioms for lattices with least element 0 and for lattices with least
element 0 and greatest element 1. It seems that Ruedin and Malliah are
the last representatives of this kind of exhaustive research in lattice theory.
Birkho [1948], Problem 7 asked what are the the consequences of
working with weakened forms of idempotency L 1and absorption L 4, namely
L22x^x=x_x;
L23x^(x_y) =x_(x^y):
Answering this problem, Matusima [1952] devised 10 systems of 6{8 axioms
for lattices, each of them containing the axioms L 2and L 3of commutativity
and associativity, plus 2{4 other axioms chosen among L 21;L22;L23and
several implications like x^y=x_y=)x=y, orx^y=y^y=)x_y=x,
etc. See also Padmanabhan [1971] and Appendix C.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.2 De ning Lattices in Terms of the Operations _and^ 13
McKenzie [1970] devised the self-dual system of axioms
L4=fL_
24;L^
24;L_
25;L^
25g;
where we have set
L_
24x_(y^(x^z)) =x;
L^
24x^(y_(x_z)) =x;
L_
25 ((y^x)_(x^z))_x=x;
L^
25 ((y_x)^(x_z))^x=x;
and used it in order to obtain further a single identity de ning lattices.
Let us prove that L4actually characterizes lattices. We shall tacitly use
the principle of duality. First we apply L_
24withy:= (y_x)^(x_z) and
z:=y_(x_z); taking into account L^
24and L^
25, we obtain
x=x_(((y_x)^(x_z))^(x^(y_(x_z))))
=x_(((y_x)^(x_z))^x) =x_x;
showing that the two operations are idempotent. Hence if we take y:=x_z
in L^
24we obtain
x=x^((x_z)_(x_z)) =x^(x_z);
therefore the two absorption laws L 4hold.
Furthermore, taking z:=xin L^
24we obtain the following variants of
absorption:
x^(y_x) =x; x_(y^x) =x:
Therefore L_
25withx:=x_yandz:=xyields
x_y= ((y^(x_y))_(x^x))_(x_y) = (y _x)_(x_y):
The latter equality implies
(y_x)^(x_y) = (y _x)^((y_x)_(x_y)) =y_x
by using L^
4, and
y_x= (x_y)_(y_x)
by interchanging xandy. It follows from the last two identities and the
second variant of absorption that
y_x= (x_y)_(y_x) = (x _y)_((y_x)^(x_y)) =x_y;
showing that the two operations are commutative.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
14 1. Semilattices and Lattices
At this point we can borrow from lattice theory the following properties:
the relation de ned by xy()x^y=xis re
exive, antisymmetric
and satis es xy()x_y=y. Besides, if yxandzxtheny_zx
because (y _z)_x=xby L_
25, wherey^x=yandx^z=z. On the
other hand xx_(y_z) by L^
4withy:=y_z, andyx_(y_z) by L^
24
withxandyinterchanged, hence zx_(z_y) =x_(y_z). Therefore
(x_y)_zx_(y_z) = (z _y)_xz_(y_x) = (x _y)_z ;
which proves the associativity of the two operations. 2
Two other characterizations of lattices generalize Theorem 1.1.
Theorem 1.2.1. (Petcu [1971]) An algebra (L;_;^)of type (2,2)is a
lattice if and only if it satis es the following two conditions, for every n
and everyn+ 1variablesx1;:::;x n;y:
(i)every two _ expressions 'and in the variables x1;:::;x ngenerate
the identity '^('_y) = ;
(ii)every two ^ expressions 'and in the variables x1;:::;x ngener-
ate the identity '_('^y) = .
Proof: IfLis a lattice then the identity '^('_y) ='holds. On the
other hand, if '; are_ expressions in the variables x1;:::;x n, then they
generate the same function by Theorem 1.1. The proof of (ii) is similar.
Conversely, suppose conditions (i) and (ii) are satis ed. Then, taking
':= :=x, we obtain the absorption laws. Therefore conditions (i)
and (ii) reduce to the following: every two _ expressions (^ expressions)
'and in the same variables generate the same function. In view of
Theorem 1.1, this implies that (L; _) and (L; ^) are semilattices. 2
Theorem 1.2.2. (Petcu [1971]) An algebra (L;_;^)of type (2,2)is a
lattice if and only if it satis es the following two conditions, for every n
and everyn+ 2variablesx1;:::;x n;y;z:
(j)if'and are_ expressions or ^ expressions in the variables
x1;:::;x n, then they generate the identity
'^('_y) = _( ^z) ;
(jj)the identity '(x;:::;x) = xholds.
Proof: IfLis a lattice then (j) and (jj) follow from Theorem 2.1 and
Lemma 1.2, respectively.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.2 De ning Lattices in Terms of the Operations _and^ 15
Conversely, suppose conditions (j) and (jj) are satis ed. Taking ':=
:=x:=zin (j) and applying (jj) twice we obtain
x^(x_y) =x_(x^x) =x_x=x
andx_(x^y) =xis obtained similarly. Now condition (j) reduces to
conditions (i) and (ii) in Theorem 2.1, therefore Lis a lattice. 2
To state the following results we need to generalize the prerequisites
used in x1. Given an algebra (L; _;^) of type (2,2), the set of _;^ terms
is the least set of words obeying the rules 1) every variable is a term, and
2) if'and are terms, then (' _ ) and (' ^ ) are terms. The functions
generated by _;^ terms are said to be _;^ polynomials ; ifLis a lattice,
they are also called lattice polynomials. Furthermore, identities are de ned
as inx1.
Lemma 1.2.1. If the algebra (L;_;^)of type (2,2)satis es L_
1andL^
1,
then every _;^ polynomial f satis es f(x;:::;x) = x.
Proof: Similar to the proof of Lemma 1.2. 2
Corollary 1.2.1. Every lattice polynomial f satis es f(x;:::;x) = x.
A class of lattices is called nitely de nable if it is de ned within the
class of all lattices by a nite set of identities, that is, if the class consists
of those lattices that satisfy a certain nite set of identities.
Theorem 1.2.3. (Padmanabhan [1968]) Every nitely de nable class of
lattices can be de ned by a single identity within the class of all lattices.
Proof: It suces to show that any set of two identities
(3) f1=g1&f2=g2
is equivalent to the single identity
(4) f1(x1;:::;x n)_f2(y1;:::;y p) =g1(x1;:::;x n)_g2(y1;:::;y p);
where the variable sets fx1;:::;x ngandfy1;:::;y pgare disjoint.
Clearly (3) implies (4). Conversely, suppose identity (4) holds. Taking
y1:=:=yp:=g1(x1;:::;x n) and using Lemma 1.1 we get
f1(x1; : : : ; x n)_g1(x1; : : : ; x n)=g1(x1; : : : ; x n)_g1(x1; : : : ; x n)=g1(x1; : : : ; x n):
Takingy1:=:=yp:=f1(x1;:::;x n) we obtain
f1(x1; : : : ; x n)=f1(x1; : : : ; x n)_f1(x1; : : : ; x n)=g1(x1; : : : ; x n)_f1(x1; : : : ; x n);
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
16 1. Semilattices and Lattices
hencef1(x1;:::;x n) =g1(x1;:::;x n) and similarly we prove that
f2(y1;:::;y p) =g2(y1;:::;y p). 2
Another major direction of research consists in looking for systems with
as few identities as possible. It seems that for lattices this trend was inau-
gurated by Sorkin and Ponticopoulos, independently of each other. Sorkin
[1962] de ned lattices by the two absorption laws L 4plus a third iden-
tity with 23 occurrences of 9 variables, while Ponticopoulos [1962] used the
idempotency laws L 1plus a third identity with 16 occurrences of 9 vari-
ables. We have already referred to the numerous three-identity systems due
to Petcu and Malliah. Another such system is
L5=fL^
24;L_
25;L^
26g;
given by McCune and Padmanabhan [1996], where we have set
L^
26 ((x_y)^(x_z))^x=x:
A class of algebras is said to be n-based if it can be de ned by a set of n
identities, also known as a basis of the class. Sorkin [1962] proved that any
class of lattices de ned by nitely many identities is 3{based. This result
was improved in the case of lattices.
Theorem 1.2.4. (Padmanabhan [1968]) Let f and g be _;^ polynomials
of n variables x1;:::;x nover an algebra (L;_;^)of type (2,2). Then L is
a lattice satisfying the identity f=gif and only if it ful ls the following
identities:
L_
27 (x^y)_y=y(which is the same axiom as L_
7),
L_
28 (((x^f)^z)_u)_v= (((g ^z)^x)_v)_((t_u)^u).
Proof: Putz:=u; x :=vin L_
28; by L_
27we get
(5) u_v=v_((t_u)^u):
Puttingt:=x^uin the above and using L_
27we have
(6) u_v=v_(u^u):
Puttingu:=x^vin the above and applying L_
27we get
(7) v=v_((x^v)^(x^v)):
Puttingv:= (u^u)^(u^u) in (6) and using (7) and L_
27we get
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.2 De ning Lattices in Terms of the Operations _and^ 17
(8) u=u^u:
Putx:=:y=:uin L_
27; by (8) we have
(9) u_u=u:
Putv:=uin (5); by (9) we get
(10) u=u_((t_u)^u):
Putv:= (t_u)^uin (5); again by (9) we have
(11) u_((t_u)^u) = (t _u)^u:
From (10) and (11) we see that
(12) u= (t_u)^u:
So, (5) reads as
(13) u_v=v_u:
Now L_
28becomes
(14) (((x^f)^z)_u)_v= (((g ^z)^x)_v)_u:
Putx:=x1=:=xn:=z. It follows from (8) and (9) via Lemma 2.1
that
(x^f(x;:::;x)) ^x=x= (g(x;:::;x) ^x)^x;
hence (14) reduces to
(15) (x_u)_v= (x_v)_u:
Therefore, by (15) and (13),
(x_u)_v= (v_x)_u= (v_u)_x;
by applying (13) twice, we get
(16) (x_u)_v=x_(u_v):
Thus (L; _) is a semilattice.
Now we take u:=vin (14) and obtain
((x^f)^z)_u= ((g^z)^x)_u;
which implies, by taking in turn u:= (x^f)^zandu:= (g^z)^x, that
(17) (x^f)^z= (g^z)^x:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
18 1. Semilattices and Lattices
Further we take x1:=:=xn:=yin (17) and using again Lemma 2.1 we
obtain (x ^y)^z= (y^z)^x. The latter identity and (8) show that (L; ^)
is a semilattice according to system fS1;S6ginx1. Since the absorption
laws L 27and (12) also hold, it follows that (L; _;^) is a lattice.
Finally we obtain the identity f=gby takingx:=z:=f_gin (17).
2
Note that equation L 28involvesn+ 5 variables, while the letters
f;g;x;z;u;v andthave 12 ocurrences. The paper by Padmanabhan [1969b]
establishes a result which resembles Theorem 2.4, except that instead of L 25
there is a similar equation with 14 occurrences of letters.
Corollary 1.2.2. (Kalman [1968]) Lattices are characterized by the iden-
tities L_
27and
L_
29 (((x^y)^z)_u)_v= (((y ^z)^x)_v)_((t_u)^u):
Proof: Characterize the class of all lattices by the identity y=y. 2
Corollary 1.2.3. Every nitely de nable class of lattices can be charac-
terized by two identities, namely L27and an identity of the form L28:
Proof: By Theorems 2.3 and 2.4. 2
Tamura [1975] devised the system
L6=fL_
27;L_
30g;
where L_
30is a slight improvement of L_
29:
L_
30 (((x^y)^z)_u)_v= (((y ^z)^x)_v)_((y_u)^u):
Padmanabhan [1972] suggested the system
L7=fL^
31;L_
32g;
where we have set
L^
31 (x_y)^z= (z^(x_x))_(z^(y_x));
L_
32 ((z^x)_(z^y))_((z^z)_(z^z)) =z:
A self-dual system of identities de ning lattices was given by Padman-
abhan [1983], namely
L8=fL_
7;L^
7;L_
33;L^
33g;
where
L_
33 (x_y)_z= (y_z)_x;
L^
33 (x^y)^z= (y^z)^x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.3 One-Based Theories 19
while in fact L_
7and L^
7can be replaced by any other variants of the absorp-
tion laws. The proof is very easy: the absorption laws imply the idempo-
tency laws (cf. L1), therefore _and^are semilattice operations according
to Padmanabhan's system fS1;S6g.
Tamura [1975] characterized lattices with 0 by the identities L_
27and
L_
34, where
L_
34 ((((0_a)^b)^c)_d)_e= (((b ^c)^a)_e)_((b_d)^d);
while the following system of two identities for the lattices with 0 and 1 is
due to Soboci nski [1979]:
L_
35 (y^(z^x))_x=x;
L_
36 ((x^(y^z))_t)_u= ((((z ^1)_0)^(x^y))_u)_((v_t)^t):
Quite recently, McCune and Padmanabhan found a new system of
axioms:
Theorem 1.2.5. The following self-dual set of two identities characterizes
lattices:
L_
62 (((x^y)_y)^(z_y))_(u^((v^y)_(y^w))) =y;
L^
62 (((x_y)^y)_(z^y))^(u_((v_y)^(y_w))) =y:
See Appendix A for a proof provided by the computer program Prover9.
1.3. One-Based Theories
From the point of view of this section, the equational theory of a class of
algebras is the collection of all equations (i.e. identities) that hold in all
members of that class. As a trivial example, the equational theory of a
class of one-element algebras consists of all identities of the relevant type
and is generated by the single identity x=y. In the other extreme of the
spectrum, the equational theory of the class of all structures of a given type
contains only equations of the form x=x. For more details the reader is
referred to the excellent survey article on this topic by Tarski [1968].
Recall that a class of algebras is said to be n-based if it can be de ned
by a set ofnidentities. Then that class is a variety and we will alternatively
say that the equational theory of that variety is n-based.
Given a nitely-based equational theory Tof algebras, it is but natural
to ask for the minimum number of equations that a basis for Tcan con-
tain, and in particular, to determine whether Thas a basis consisting of
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
20 1. Semilattices and Lattices
a single identity. The answer is well known for group-like systems. Every
nitely-based theory of groups (or loops) is always one-based (results due
to Higman and Neumann [1952], Tarski [1968], Padmanabhan [1969a]).
These algebras admit cancellation laws and have quasi-group properties,
and these play a crucial role in constructing a single axiom for group-like
theories. However, lattices neither admit any cancellation laws nor they
enjoy any meaningful quasi-group property. In this sense, the equational
theory of lattices is \far removed" from that of group theory. In view of
these intuitive observations, it was widely believed – till 1967 – that the
equational theory of lattices may not be one-based and, in fact, not even
de nable by any set of identities of the form f(x;x1;:::;x n) =x, which is,
prima facie, essential for any potential one-based theory. It was McKenzie
[1970] who rst published the theorem that the variety of all lattices can,
indeed, be de ned by such \absorption laws" and that the equational the-
ory is, in fact, one-based. McKenzie also mentioned in the same publication
that no other variety of lattices (except the variety of singletons, de ned
byx=y) is one-based. Hence for the nitely based lattice varieties \two"
(as proved in Theorem 2.4) is best possible.
In this section we regard semilattices as algebras of type (2) and lattices
as algebras of type (2,2) (disjunction and conjunction). We will show that
while semilattices cannot be de ned by a single axiom, lattices can be so
de ned.
We need to introduce some terminlogy. By an absorption identity we
mean an identity of the form f(x;x1;:::;x n) =x. An identity f=g
is called regular if the sets of variables occurring on the two sides of the
equation are the same.
Lemma 1.3.1. Any system of identities de ning a class of lattices contains
an absorption identity.
Proof: A non-absorption identity is of the form f=g,where neither
fnorgis a variable and hence both fandgcontain at least one of the
operation symbols _;^. Now take the two-element set f0;1gand de ne
x_y=x^y= 0 for all xandy. This algebra will satisfy all the non-
absorption identities but 1 _1 = 0 6= 1 and hence is not idempotent. 2
Lemma 1.3.2. Any identity valid in a semilattice is regular.
Proof: The axioms S 1;S2;S3are regular and regularity is preserved under
equational consequences (by substitutions). 2
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.3 One-Based Theories 21
Theorem 1.3.1. (Potts [1965]) The variety of all semilattices cannot be
de ned by a single identity.
Proof: It follows from Theorems 3.1 and 3.2 that a potential single iden-
tity for the class of all semilattices must be of the form f(x;:::;x) = x.
Supposexoccursn+1 times on the left-hand side. If n>1 take the additive
group (Z n;+) and de ne x_y=x+y. Thenf(x;:::;x) = (n + 1)x =x,
but this model of the identity f=xis not idempotent. If n= 1 the equa-
tion isxx=xand it is very easy to produce a three-element groupoid
which is idempotent but not associative. Contradiction. 2
The next theorem uses the concept of J onsson term, also called majority
polynomial. This means a ternary polynomial psatisfying the identities
(1) p(x;x;z ) =p(x;y;x) = p(y;x;x) = x:
For instance, lattices admit the majority polynomial
p(x;y;z ) = (x ^y)_(y^z)_(x^z):
Theorem 3.2 below provides a very simple identity to show that any
nitely-based variety of algebras that admits a J onsson term and is de n-
able by absorption identities, is one-based. Our knowledge of this result
is due to the paper by McKenzie [1970] and forms a part of Theorem 1.2
stated there without proof.
Lemma 1.3.3. (Padmanabhan [1977]) Let T be an equational theory with
a majority polynomial p. For arbitrary polynomials f and g, the validity of
two identities f=xandg=xin T is equivalent to p(f;g;y ) =x, where y
is a variable not occurring in f or g.
Proof: Clearlyf=xandg=xtogether imply p(f;g;y ) =x. To get
the converse, substitute y:=fto derivef=xandy:=gto deriveg=x.
2
Theorem 1.3.2. (Padmanabhan [1977]) Let T be an equational theory
which is de ned by nitely many absorption identities and has a majority
polynomial. Then T is one-based.
Proof: By Lemma 3.3 we can assume that Thas a basis of the form
f=yand the three identities (1) which de ne a majority polynomial p.
Now consider
(2) p(p(x;y;y );u;p(p(x;y;y );f;z )) =y;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
22 1. Semilattices and Lattices
wherezanduare variables not occurring in f. Certainly Timplies the
identity (2). Conversely, let us prove that (2) implies (1) and f=y.
Putz:=p(p(x;y;y );f;w ) in (2). Since p(p(x;y;y );f;p(p(x;y;y );
f;w)) =yby (2), we obtain
(3) p(p(x;y;y );u;y ) =y:
Puttingx:=p(a;y;y ) in (3) and noting that p(p(a;y;y );y;y ) =yagain by
(3), we get
(4) y=p(p(p(a;y;y );y;y );u;y ) =p(y;u;y ):
Putz:=p(x;y;y ) in (2). By two successive applications of (4) we have
(5)y=p(p(x;y;y );u;p(p(x;y;y );f;p(x;y;y )))
=p(p(x;y;y );u;p(x;y;y )) =p(x;y;y ):
Thus (2) becomes p(y;u;p(y;f;z )) =y, which, by substituting u:=
p(y;f;z ) and using (5), yields
(6) y=p(y;p(y;f;z );p(y;f;z )) =p(y;f;z ):
Finally,z:=fin (6) implies, by (5) again, the identity
(7) f=y;
which, in turn, reduces (6) to
(8) p(y;y;z ) =y:
Thuspis a majority polynomial by (4), (5) and (8), and moreover, we
have the identity f=y. 2
As was mentioned in the beginning, McKenzie rst constructed a single-
equation basis for lattices, involving 34 variables. However, starting from
the four absorption identities given by McKenzie [1970] ( 1;2;3;4on
page 27) and using Lemma 3.3 we get an identity f=ywith ve variables,
and applying Theorem 3.3 to this identity and the three identities (1), we
obtain a single axiom for lattices in only seven variables. More generally,
ifKis an equational class de ned by nabsorption identities involving at
mostkvariables and if Kadmits a majority polynomial, then Khas a
one-basis involving at most k+ 2 + log2nvariables (G.M. Bergamn, Reno
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.3 One-Based Theories 23
AMS Conference). On the other hand, Gr atzer [1998] (cf. Problem 17) asks
for a short identity de ning lattices. The following table reports numerical
characteristics of several single axioms de ning lattices:
Reference Variables Length
McKenzie [1970] 34 300,000
Padmanabhan [1977] 7 243
McCune and Padmanabhan [1996a,b] 7 79
Vero [2001] 8 77
McCune, Padmanabhan and Vero [2003] 8 29
Thus the latter paper provides the shortest known single axiom for the
equational theory of all lattices, having length 29 and 8 variables:
L^
36 (((y_x)^x)_(((z ^(x_x ))_(u^x ))^v))^(w_((s_x)^(x_t))) =x;
the length being understood as the number of occurrences of variables and
operation symbols.
This single-identity for lattices was found with the aid of a computer
program called Otter (Organized Tools and Techniques for Ecient Re-
search), devised by McCune; cf. McCune and Padmanabhan [1996a]. The
search strategy was the following:
{ generate candidates of the form f=x;
{ eliminate candidates that are not lattices identities by incorporating
certain equational lters in the program;
{ for each candidate, try either to nd a small nite nonlattice model
of it, or to derive from it the standard system L1.
Let us explain the elimination of candidates by an actual example en-
countered by the machine in the process of discovering a single axiom for
lattices. Otter discovered the identity
(L*) (y_((x_z)^(z_x)))^(((x^t)_x)_((u^x)_(x^w))) =x
and it is easy to verify that it is valid in all lattices. However it is not
strong enough to derive all the axioms for lattices. The reason is that we
can \parse" the above identity and nd a stronger class of lattice identities
which is well known to be inadequate for de ning lattices. Here is the
parsing process:
(x^t)_x=x;
x_((u^x)_(x^w)) =x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
24 1. Semilattices and Lattices
(y_((x_z)^(z_x)))^x=x:
Although these three identities do imply the single identity (L*), there is a
non-lattice model satisfying them, hence (L*) as well, therefore (L*) is not
a single identity for lattice theory. Here is one such model.
Take the ve-element lattice f0;c;a;b; 1gwith 0<c=a^b<a_b= 1.
Re-de nea^b=b^a= 0, otherwise let _and^be the lattice operations.
Since the join operation is a semilattice operation, the relation xyde ned
byx_y=yis a partial order. Also, this algebra satis es all the two-
variable lattice laws, because the subalgebra generated by any two elements
is, indeed, a lattice. Therefore
(x^t)_x=x;
x_((u^x)_(x^w)) = ((x _(u^x))_(x^w) =x_(x^w) =x;
(y_((x_z)^(z_x)))^x= (y_(x_z))^x=x:
However (a ^b)^c= 0^c= 0, whilea^(b^c) =a^c=c. Thus identity
(L*) is eliminated. Such counter-examples are incorporated in the software,
so that it can lter out the non-lattice identities automatically.
The program was run on several hundred processors, usualy in jobs of
10-20 hours, over a period of several weeks. Two short single-identities for
lattices were found, namely L^
36and
L^
37 (((y_x)^x)_(((z ^(x_x ))_(u^x ))^v))^(((w_x)^(s_x))_t) =x;
while for many shorter equations and equations with fewer variables, nei-
ther a proof of L1nor a nonlattice model could be found (caution: only a
nonlattice model of an equation eliminates it from the list of candidates!).
The program consists in fact of several programs with specialized jobs.
The program for proof searching, called Otter, derived L1from L^
36in more
than 250 steps. Then Otter was used to obtain L4, which it did in about
170 steps. Later on, L. Wos used various methods to simplify the Otter
proof and obtained a proof in 50 steps. Each step uses paramodulation,
an inference rule that combines variable instantiation (or uni cation) and
equality substitution into one step.
We give below a variant of the latter proof. We indicate paramodula-
tion in the following form: \take <substitution S1> ini, then use jwith
<substitution S2>". More exactly, let i1=i2andj1=j2be the equations
iandj, respectively. Let i0
1=i0
2andj0
1=j0
2be the equations S1(i) and
S2(j), respectively. The resulting equation is i00
1=i0
2, wherei00
1is obtained
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.3 One-Based Theories 25
fromi0
1by replacing the subterm(s) j0
1byj0
2(ori00
1=i00
2, wherei00
2is ob-
tained from i0
2by the same transformation). For instance, \Take y:=x^y
inx^(x_y) =x, then use x_(x^y) =x" produces the equation x^x=x;
here S2 is the identity.
1. (((y _x)^x)_(((z^(x_x))_(u^x))^v))^(w_((s_x)^(x_t))) =x:
This is axiom L^
36.
2. (((x _y)^y)_(y_y))^(z_((u_y)^(y_v))) =y:
Sety_y=Y. Takex:=y; y :=x; z :=y_Y; u := (z^(Y_Y))_(u^
Y); v:=w_((s_Y)^(Y_t)); w :=z; s :=u; t:=vin 1, then use 1
withx:=Y; v :=y.
3. (((x _(y_y))^(y_y))_((y_y)_(y_y)))^(z_y) =y_y:
Takey:=y_y; u := (x _y)^y; v := (u _y)^(y_v) in 2, then use 2
withz:=y_y.
4. (((x _y)^y)_(((y_y)_(z^y))^u))^(v_((w_y)^(y_t))) =y:
Takex:=y; y:=x; z := ((x _(y_y))^(y_y))_((y_y)_(y_y)); u :=
z; v:=u; w :=v; s:=win 1, then use 3 with z:=y.
5. (((x _Y)^Y)_(Y_Y))^(v_y) =Y, whereY= ((y_y)_(z^y))^u.
Takey:=Y; z :=v; u := (x_y)^y; v := (w _y)^(y_t) in 2, then use
4 withv:=Y.
6. (((x _y)^y)_(((((y _y)_(z^y))^u)_(v^y))^w))^(t_((s_
y)^(y_r))) =y.
Set ((y _y)_(z^y))^u=Y. Takex:=y; y :=x; z := ((x _Y)^Y)_
(Y_Y); u:=v; v:=w; w :=t; t:=rin 1, then use 5 with v:=y.
7. (((x _y)^y)_(z^y))^(u_((v_y)^(y_w))) =y.
Setz^y=Y. Takeu:=Y; v := (((Y _Y)_(z^Y))^u)_(v^Y); w :=
t_((s_Y)^(Y_r)); t :=u; s :=v; r :=win 6, then use 6 with
x:=y_y; y:=Y; w :=y.
8. (((x _(y^z))^(y^z))_(u^(y^z)))^(v_z) =y^z.
Takey:=y^z; z:=u; u :=v; v:= (x_z)^z; w := (v_z)^(z_w) in
7, then use 7 with y:=z; z:=y; u :=y^z.
9. (((x _y)^y)_(((z^y)_(u^y))^v))^(w_((t_y)^(y_s))) =y.
Takex:=y; y:=x; z := ((x _(z^y))^(z^y))_(u^(z^y)); s :=t; t:=s
in 1, then use 8 with y:=z; z:=y; v:=y.
10. (((x _y)^y)_y)^(z_((u_y)^(y_v))) =y.
Takez:=x_y; u := (z^y)_(u^y); v:=w_((t_y)^(y_s)); w :=
z; t:=u; s:=vin 9, then use 9 with v:=y.
11. (((x _y)^y)_(((z^y)_(u^y))^v))^(w_y) =y.
Taket:= (x_y)^y; s:= (u_y)^(y_v) in 9, then use 10 with z:=y.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
26 1. Semilattices and Lattices
12. (((x _y)^y)_(z^y))^(u_y) =y.
Takev:= (x _y)^y; w := (x _y)^(y_v) in 7, then use 10 with
z:=y; u :=x.
13. (((x _y)^y)_(y_y))^(z_y) =y.
Takeu:= (x _y)^y; v := (u _y)^(y_v) in 2, then use 10 with
z:=y; u :=x.
14. (x _(y^(x_x)))^(z_((u_(x_x))^((x_x)_v))) =x_x.
Takex:= (x_x)^x; y :=x_x; z :=y; u :=z; v:=u; w :=vin 7, then
use 13 with y:=x; z:=x.
15. (x _x)^(y_((z_(x_x))^((x_x)_u))) =x_x.
Takey:= ((x _x)^x)_(x_x); z :=y; u :=z; v:=uin 14, then use 13
withy:=x; z:=x.
16. (((x _y)^y)_((z^y)_(z^y)))^(u_y) =y.
SetX=z^y. Takeu:=z; v:=y_((z_(X_X))^((X_x)_u)); w :=u;
in 11, then use 15 with x:=X; w :=u.
17. ((x ^y)_(x^y))^(z_y) = (x ^y)_(x^y).
Takex:=x^y; y :=z; z := (x_y)^y; u :=yin 15, then use 16 with
z:=x; u := (x^y)_(x^y).
18. ((x _y)^y)_((x_y)^y) =y.
Takex:=x_yin 17, then use 12 with z:=x_y; u :=z.
19. (x ^y)^(z_y) =x^y.
Takey:=x; z :=y; u :=x_(x^y); v :=zin 8, then use 18 with
y:=x^y; z:=x.
20.x^(y_x) =x.
Takex:=y; y:=x; z :=y_x; u :=yin 12, then use 18 with x:=y; y:=
x.
21.x^(y_((z_x)^(x_u))) =x.
Takex:=y; y :=x; z :=y_x; u :=y; v :=z; w :=uin 7, then use 18
withx:=y; y:=x; z :=y.
22. (((x _y)^y)_((z^y)_(u^y)))^(v_y) =y.
Takev:=x_((z^y)_(u^y)); w :=vin 11, then use 20 with x:=
(z^y)_(u^y); y:=x.
23. ((x ^y)_(z^y))^(u_y) = (x ^y)_(z^y).
Takex:= (x^y)_(z^y); y:=u; z := (x_y)^y; u :=yin 21, then use
22 withz:=x; u :=z; v:= (x^y)_(z^y).
24. ((x _y)^y)_(z^y) =y.
Takex:=x_yin 23, then use 12.
25.x^(x_y) =x.
Takey:= (x _(x_y))^(x_y); u :=yin 21, then use 24 with y:=
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.3 One-Based Theories 27
x_y; z =z_x.
26. ((x _y)^y)_((z^y)_(u^y)) =y.
Takev:= ((x _y)^y)_((z^y)_(u^y)) in 22, then use 25 with x:=
((x_y)^y)_((z^y)_(u^y)).
27. ((x _y)^y)_y=y.
Takez:= ((x _y)^y)_yin 10, then use 25 with x:= ((x _y)^y)_y; y :=
(u_y)^(y_v).
28.x^(y_(x_z)) =x.
Takey:=x_z; z:=yin 19, then use twice 25 with y:=z.
29. (x ^x)_x=x.
Takex:= (x_x)^x; y :=xin 27, then use 27 with y:=x.
30.x^(y_(x^(x_z))) =x.
Takez:= (y_x)^x; u :=zin 21, then use 27 with x:=y; y:=x.
31.x^x=x.
Takey:= (y_x)^xin 20, then use 27 with x:=y; y:=x.
32.x_x=x.
By 29 via 31.
33.x^((y_x)^(x_z))) =x.
Takey:= (y _x)^(x_z); z :=y; u :=zin 21, then use 32 with
x:= (y_x)^(x_z).
34. (x _y)^y=y.
Takez:=x_yin 24, then use 32 with x:= (x_y)^y.
35.x^(y^x) =y^x.
Takex:= (x_x)^x; y :=y^xin 34, then use 24 with y:=x; z :=y.
36. (x _(((y^x)_(z^x))^u))^(v_x) =x.
Takex:=y; y:=x; z :=y; u :=z; v:=u; w :=vin 11, then use 34 with
x:=y; y:=x.
37. ((x _y)^(y_z))^y=y^((x_y)^(y_z)).
Takex:= (x_y)^(y_z) in 35, then use 33 with x:=y; y:=x; z:=x.
38. (x _(x^y))^(z_x) =x.
Takey:=y_x; u :=y; v:=zin 36, then use 24 with x:=y; y:=x.
39.x_(((y^x)_(z^x))^u) =x.
Takev:=x_(((y^x)_(z^x))^u) in 36, then use 25 with x:=x_(((y^
x)_(z^x))^u); y :=x.
40. ((x _y)^(y_z))^y=y.
Identity 37 reduces to 40 by 33 with x:=y; y:=x.
41. (((x ^y)_(z^y))_(((x^y)_(z^y))^u))^y= (x^y)_(z^y).
Takex:= (x^y)_(z^y); y:=u; z := (x_y)^yin 38, then use 26 with
z:=x; u :=z.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
28 1. Semilattices and Lattices
42. (x ^y)^x=x^y.
Takex:=x^y; y:=x; z:=xin 33, then use 38 with z:=x^y.
43.x_(((y^x)_x)^z) =x.
Takez:=y_x; u :=zin 39, then use 34 with x:=y; y:=x.
44.x_((y^x)^z) =x.
Takez:=y; u :=zin 39, then use 32 with x:=y^x.
45. ((x ^y)_y)^y= (x^y)_y.
Takex:= (x ^y)_yin 30, then use 43 with x:=y; y :=x; z :=
((x^y)_y)_z.
46.x_(y^(z^x)) =x.
Takey:=z; z:=y^(z^x) in 44, then use 35 with x:=z^x.
47. (x ^y)_y=y.
Identity 45 reduces to 47 by 34 with x:=x^y.
48.x_(y^(x^z)) =x.
Takez:=x^zin 46, then use 42 with y:=z.
49. ((x ^y)_(z^y))_y=y.
Takex:= ((x ^y)_(z^y))_(((x^y)_(z^y))^u) in 47, then use 41.
50. ((x ^y)_(y^z))_y=y.
Takez:=y^zin 49, then use 42 with x:=y; y:=z.
Finally note that identities 28, 40, 48 and 50 are L^
24;L^
25;L_
24and L_
25,
respectively. 2
McCune has now created Prover9 { a new software, an improved version
of Otter, and he recommends using only Prover9.
So the (improper) variety Lof all lattices can be characterized by a
single axiom. Likewise, the trivial variety Tof one-element lattices is char-
acterized by the axiom x=y. These simple remarks cannot be improved:
Theorem 1.3.3. (McKenzie [1970]) No non-trivial proper variety of lat-
tices can be de ned by a single identity.
Proof: Suppose that such a variety Kis characterized by a single identity,
which, by Lemma 3.1, is of the form f(x;x1;:::;x n) =x. Take a non-trivial
latticeL2K. Then there exist o;u2Lsatisfyingo <u and the identity
f=xholds in the sublattice fo;ug ofL, therefore it is valid in every
two-element lattice.
Now de ne f0(x;x1;:::;x n) =f(x;z;:::;z ), wherez=x^x1^:::^xn,
andf1(x;x1;:::;x n) =f(x;u;:::;u), where u=x_x1_:::_xn. Then
bothf0=xandf1=xare valid in any two-element lattice.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.4 De ning Lattices by Other Tools 29
On the other hand z < x < u because the variables x;x1;:::;x nare
distinct. Since lattice polynomials are isotone, the inequalities
(9) f0ff1
hold in any lattice.
The validity of f0=xin the lattice fz;xg implies that f0=zorf0=x,
where in fact x;x1;:::;x nare arbitrary. In other words we have f0=z
orf0=xin every lattice. The former alternative applied to the lattice
fz;xg yieldsx=z, a contradiction. Therefore f0=xin every lattice and
similarlyf1=xin every lattice. This transforms (9) into xfx, that
is, the identity f=xholds in any lattice, which contradicts the initial
assumption. 2
Corollary 1.3.1. A variety Kof lattices can be de ned by a single axiom
i either K=T (the trivial variety of one-element lattices), or K=L (the
variety of all lattices).
Proof: This summarizes Theorem 3.3 and the comments preceding it.
2
In particular the variety Dof all distributive lattices is not one-based.
However we know by Corollary 2.3 that every nitely-based variety of lat-
tices is two-based. We can apply the above idea of absorption law to prove
that several varieties of enriched lattices obtained by adjoining 0 or 1 are
one-based. This is because the property of an element being 0 or 1 can
be captured by an absorption law: x_0 =xandx^1 =x, respectively.
This idea can be streched further. An algebra (L; _;^;0) is a comple-
mented lattice if and only if it is a lattice satisfying the absorption laws
x_(y^y0) =xandx^(y_y0) =x. Summarizing the previous results
and anticipating results of later sections, we have the following: the vari-
eties of all lattices, trivial lattices, lattices with least element 0, lattices with
greatest element 1, bounded lattices (with 0and1),`-groups and Boolean
algebras are one-based, while the varieties of semilattices, quasilattices and
nitely-based lattices are two-based but not one-based.
1.4. De ning Lattices by Other Tools
In this section we present de nitions of lattices using other tools than the
orderor the operations _;^, namely segments or lattice betweenness for
lattices with least element 0, K-segments for arbitrary lattices, a quaternary
operation for nitely de nable equational classes of lattices, and a partially
de ned ternary operation for lattices wirh 0 and 1.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
30 1. Semilattices and Lattices
For every two elements a;bof a lattice L, the segment [a;b] is de ned
by
(1) [a;b] = fx2Lja^bxa_bg:
Ifab, the segment [a;b] reduces to the interval
(2) [a;b] = fx2Ljaxbg:
Theorem 1.4.1. (Nishig ori [1954])y )An algebra (L;_;^;0)of type
(2,2,0) is a lattice with least element 0if and only if with each couple
(x;y)2L2is associated a nonempty subset [x;y]ofLsuch that the fol-
lowing conditions are ful lled: for every x;y;u;v 2L,
L38 [x;x] = fxg;
L39 [0;x]\[0;y] = [0;x ^y];
L40 [0;x][0;x_y];
L41 [0;y][0;x_y];
L42 [0;x][0;v] & [0;y ][0;v] =)[0;x_y][0;v];
L43 [x;y][u;v]() [0;u]\[0;v][0;x]\[0;y] & [0;x _y]
[0;u_v]:
)When this holds, the \segments" [a;b]coincide with (1)and the order
relation of the lattice is given by
(3) xy() [0;x][0;y]:
Proof: Given a lattice Lwith 0, de nition (1) immediately implies (3)
and L 38 L43.
Conversely, suppose L 38 L43hold and de ne the relation by (3).
Then is re
exive and transitive, and
(4) x^yx&x^yy;
(5) xx_y&yx_y;
(6) xv&yv=)x_yv ;
while L 43can be written in the form
(7) [x;y][u;v]()u^vx^y&x_yu_v:
yL42is a weakening of the original axiom.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.4 De ning Lattices by Other Tools 31
Then
(8) vx&vy=)vx^y;
because the hypothesis implies [0;v ][0;x]\[0;y] = [0;x ^y].
Takingy:=v:=xin (4), (8), (5) and (6), we obtain
(9) x^xxx^x&xx_xx:
It follows from L 37and L 38that
f0g \ [0;x] = [0; 0]\[0;x] = [0; 0^x]6=?;
hence 0 2[0;x], therefore [0; 0] =f0g [0;x], that is,
(10) 0x:
It remains to prove (1) and antisymmetry.
Taking in (7) y:=xand using L 38and (9), we infer
x2[u;v]() [x;x][u;v]()u^vx^x&x_xu_v
=)u^vx&xu_v=)u^vx^x&x_xu_v()x2[u;v];
therefore we get (1) in the form
x2[u;v]()u^vx&xu_v:
Finally it follows from (9) that
xu=)x_xxuu_u;
ux=)u^uuxx^x;
hence
xu&ux=)x_xu_u&u^ux^x:
On the other hand we infer from (7) with y:=xandv:=uthat
x=u() fxg fug () u^ux^x&x_xu_u;
therefore
(11) xu&uz=)x=y:
2
Smiley and Transue [1943] have worked with the concept of lattice be-
tweenness, which is the ternary relation de ned by
(12) axb() (a^x)_(x^b) =x= (a_x)^(x_b):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
32 1. Semilattices and Lattices
This concept originates in the theory of metric lattices, initiated by
Glivenko. In a lattice endowed with a distance function d, an element
xis said to be between two elements a;bifd(a;x) +d(x;b) =d(a;b).
Glivenko proved that this happens if and only if the above property axb
holds. But de nition (12) makes sense in any lattice and the following easy
consequences of (12) have been considered by Smiley and Transue:
L44abc()cba;
L45abc&acb()b=c;
L46abc&adb=)dbc;
and if the lattice has least element 0, then
(13) xy() 0xy;
L47 0bc& 0dc &bxd=)0xc;
L48 (a_b)a(a^b);(a_b)b(a^b); a(a_b)b; a(a ^b)b;0(a^b)(a_b);
L49x^(p_c) = (p ^x)_(x^c) &x_(p^c) = (p _x)^(x_c) =)
(pbc&pdc&bxd=)pxc):
Property L 44is immediate. If abc&acbthena^b(a_c)^(c_b) =c,
henceb= (a^b)_(b^c)cand similarly cb, henceb=c, thus proving
L45. Ifabc&adbthen from (a _d)^(d_b) =dwe infer (a _d)^b=b^d,
then
b= (a^b)_(b^c)((a_d)^b)_(b^c) = (b ^d)_(b^c)b;
henceb= (d^b)_(b^c) and similarly b= (d_b)^(b_c). Therefore L 46
holds.
One checks readily that 0xy ()x=x^y, which proves (13). Now L 47
reads
bc&dc&bxd=)xc
and in fact the hypotheses imply
x= (b^x)_(x^d)b_dc:
Checking L 48is routine. To prove L 49note that
pbc=)bp_c; pdc =)dp_c; bxd =)xb_d;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.4 De ning Lattices by Other Tools 33
hencexp_c, andxp^cby duality. Therefore x=x^(p_c) =
(p^x)_(x^c) andx=x_(p^c) = (p _x)^(x_c).
Smiley and Transue (op. cit.) have proved a theorem which is slightly
more general than the following
Theorem 1.4.2. )An algebra (L;_;^;0)of type (2,2,0) is a lattice with
rst element 0if and only if it is endowed with a ternary relation satisfying
L44 L49.
)When this holds, the \lattice betweenness" coincides with (12), while
the order relation of the lattice is given by (13).
See also Blumenthal and Bumcrot [1962].
Starting from lattice betweenness, Kolibiar [1958] introduced a concept
which we will call K-segment. Given a lattice Land two elements a;b2L,
let
(14) B(a;b) = fx2Ljaxbg
be the set of elements that are lattice-between aandb. Note that
(15) a; b; a ^b; a_b2B(a;b);
(16) B(a;b)[a;b]B(a^b;a_b):
Let us refer to a subset FLsatisfyinga;b2F=)B(a;b)2Fas a
lattice-convex set. LetX7!Xdenote the closure operator associated with
the Moore family of lattice-convex sets. Then
(17) B(a;b) = [a;b] :
For ifx;y2[a;b] thena^bx^yx_ya_b, which, using (16),
impliesB(x;y)[x;y][a;b]. Therefore [a;b] is a lattice-convex set which
includesB(a;b). Suppose B(a;b)F, whereFis a lattice-convex set. If
x2[a;b] thenx2B(a^b;a_b) by (16); but a^b;a_b2Fby (15),
thereforex2F. This completes the proof of (17).
If the setB(a;b) is closed, then we put
(18) K(a;b) =B(a;b) = [a;b]
and refer to this set as the K-segment determined by (a;b); otherwise the
coupleK(a;b) does not determine a K-segment.
Kolibiar (op. cit.) has considered the following properties of segments,
lattice betweenness and K-segments:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
34 1. Semilattices and Lattices
L50 [a;b]\[b;c]\[c;a]6=?;
L51axb() [a;x]\[b;x] = fxg;
L52 every three elements a;b;c are contained in a K-segment;
L53 with each K-segmentK(a;b) is associated an \oriented" K-
segmentK(o;u), such that
(i)K(a;b) =K(o;u), and
(ii) for every two oriented K-segmentsK(o;u);K (o0;u0),
ifK(o;u) K(o0;u0) and theK-oriented segment
K(o0;u) exists, then o2K(o0;u).
Property L 50follows from
[a;b]\[b;c]\[c;a] = [(a ^b)_(b^c)_(c^a);(a_b)^(b_c)^(c_a)];
and this also proves L 52, because every segment is closed by (17).
Further, note that since
(19) (a^x)_(b^x)x(a_x)^(b_x);
it follows by comparison with (12) that in fact
(20) axb() (a^x)_(b^x) = (a _x)^(b_x):
On the other hand,
(21) [a;x]\[b;x] = [(a ^x)_(b^x);(a_x)^(b_x)];
hence (20) says that axbholds if and only if the segment (21) is a singleton,
and in view of (19) this singleton is fxg, thus proving L 51.
To prove L 53note that if outhen
oxu() (o_x)^(x_u) =x= (o^x)_(x^u)
()o_x=x=x^u()oxu:
In other words, ou=)B(o;u) = [o;u] and we can take as oriented
K-segments, the segments [o;u] with ou. So, given K(a;b), we have
K(a;b) =K(o;u) with
(22) o=a^b; u =a_b:
Besides, a stronger form of L 53(ii) holds: if K(o;u) andK(o0;u0) are ori-
entedK-segments satisfying K(o;u) K(o0;u0), theno0ouu0,
showing that the oriented K-segmentK(o0;u) exists and o2K(o0;u).
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.4 De ning Lattices by Other Tools 35
Theorem 1.4.3. (Kolibiar [1958]) Let L be a set endowed with a function
[ ] :L2 ! P (L), a partial function K:L2 ! P (L)and a ternary relation
L3; letaxbstand for (a;x;b) 2 . Then L is a lattice (L;_;^;)in
which relations (1), (12), (14), (18) and(22)hold if and only if it satis es
L50 L53.
We omit the more elaborated rest of the proof. See also Hedlikov a and
Katri~ n ak [1991] and Plo s cica [1996].
Theorem 1.4.4. LetKbe the class of lattices de ned within the class of
all lattices by an identity f=g, wherefandgare lattice polynomials. Let
(L;_;^)be an algebra of type (2,2)and de ne
(23) q(a;b;c;d) = ((a _b)^c)_(a^d):
Further let Qbe the equation obtained from L_
28by the transformations
(24) a_b=q(b;a;a;b);
(25) a^b=q(b;b;a;a):
Then (L;_;^)2Kif and only if (L;q)satis es Q and
L54q(a;x;q (a;a;z;t);a) = a:
Proof: In view of Theorem 2.4, we have to prove that (L; _;^) satis es
L_
27and L_
28if and only if (L;q ) satis es L 54andQ.
Suppose (L; _;^) satis es L_
27and L_
28. Then _and^coincide with the
operations de ned by (24) and (25) because
q(b;a;a;b) = ((b _a)^a)_(b^b) =a_b;
q(b;b;a;a) = ((b _b)^a)_(b^a) =a^b:
ThereforeQfollows from L_
28, while L 54is ful lled because
q(a;x;q (a;a;z;t);a) = ((a _x)^(q(a;a;z;t)) _(a^a)
= ((a_x)^(((a_a)^z)_(a^t)))_a= ((a_x)^((a^z)_(a^t)))_a
= ((a^z)_(a^t))_a=a:
Conversely, suppose (L;q ) satis esQand L 54. Then it follows from (24)
and (25) that (L; _;^) satis es L_
28and also L_
27, because
(x^y)_y=q(y;x^y;x^y;y) =q(y;x^y;q(y;y;x;x);y ) =y
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
36 1. Semilattices and Lattices
by L54witha:=y; x:=x^y; z:=t:=x. 2
Corollary 1.4.1. (L;_;^)is a lattice if and only if (L;q)satis es L54and
the identity Q0obtained from L_
29by the transformation (24), (25).
Proof: From Theorem 4.4 and Corollary 2.2. 2
Corollary 1.4.2. Every nitely de nable class of lattices can be charac-
terized in terms of a quaternary operation.
Proof: By Theorems 2.3 and 4.4. 2
It was shown by D. Kelly and Padmanabhan [1989] that lattices can-
not be de ned in terms of a ternary operation alone, because a ternary
polynomial cannot express both the join and the meet.
However in a bounded lattice, that is, a lattice with 0 and 1, the ternary
operation
(26) t(a;b;c) = (a _c)^(b_(a^c))
yields the lattice operations via formulae
(27) a^c=t(a;0;c); a_c=t(a;1;c):
Martin [1965] characterized lattices by the axioms
L55t(a;b;c) =t(c;b;a);
L56t(0;1;a) =a;
L57t(1;0;a) =a;
L58t(a;1;t(b; 1;c)) =t(c;1;t(b; 1;a));
L59t(a;0;t(b; 0;c)) =t(c;0;t(b; 0;a));
L60t(a;0;t(a; 1;b)) =a;
L61t(a;1;t(a; 0;b)) =a;
to the e ect that in every lattice the operations (26) satisfy properties L 55
L61and conversely, every algebra (L;t) of type (3) which ful ls these axioms
becomes a bounded lattice (L; _;^;0;1) with respect to the operations (26).
See also the operation s(x;y;z ) =t(y;x;z ) in Chapter 2, x2, and the median
operation in Chapter 3, x3.
Kolibiar [1956a] characterized bounded lattices in terms of the partially
de ned ternary operation
(28)<a;b;c> = (a^b)_(b^c)_(c^a) = (a _b)^(b_c)^(c_a);
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
1.4 De ning Lattices by Other Tools 37
where<a;b;c> is not de ned if the second equality in (28) does not hold,
and which satis es certain axioms. Conversely, his axioms imply only a
weaker form of (28), namely
(a^b)_(b^c)_(c^a)<a;b;c> (a_b)^(b_c)^(c_a):
Katri~ n ak [1961] has shown that two operations <> and<>0satisfying the
axioms need not coincide even if they have the same domain of de nition.
Other problems concern self-dual varieties of lattices. A variety of lat-
tices is called self-dual provided the principle of duality holds in all of its
members. For instance, the varieties of all lattices, modular lattices, dis-
tributive lattices and Boolean algebras (cf. next chapters) are self-dual.
Clearly every self-dual variety can be de ned by a self-dual system of ax-
ioms. It is natural to ask whether a given nitely-based self-dual variety
of lattices can be de ned by a single self-dual axiom relative to the class
Lof all lattices. D. Kelly and Padmanabhan [2002] proved that there are
in nitely many varieties for which the answer is \yes" and in nitely many
varieties which don't have such a single-axiom characterization. Let us say
that these varieties are of the rst kind and of the second kind, respectively.
On the other hand, a self-dual system of axioms need not be independent
and an independent system need not be self-dual. Therefore it is natural
to look for irredundant (i.e., independent) self-dual bases for nitely based
self-dual lattice varieties. D. Kelly and Padmanabhan [2004] proved the
following result. Associate with every nitely based self-dual lattice variety
Va numbern0de ned as follows: n0= 1 for T,n0= 2 for L,n0= 3 ifV
is of the rst kind, and n0= 4 if Vis of the second kind. Then for every
nitely based self-dual variety Vand everynn0there is an irredundant
self-dual basis with nidentities de ning V.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
This page intentionally left blankThis page intentionally left blank
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Chapter 2
Modular Lattices
Modular lattices were introduced by Dedekind [1897] as an abstract charac-
terization of the lattice of normal subgroups of a group; see Birkho [1948]
for historic references. Modular lattices have also a strong connection with
projective planes. Their systematic study was begun by Ore[1935].
In this chapter, after a short introduction of modular lattices as a sub-
class of the class of all lattices, we survey direct de nitions of modular
lattices, either in terms of disjunction and conjunction, or by other tools.
The last section is devoted to the construction of self-dual equational bases
for varieties of modular lattices. The nal result is that any nitely-based
self-dual variety of modular lattices can be de ned as an equational class
of lattices satisfying a single identity. This is a special case of a most
general result on the so-called p-modular lattices, proved by D. Kelly and
Padmanabhan [2002].
2.1. Modular Lattices within Lattices
A lattice (L; _;^) is said to be modular provided
(1) xz=)x_(y^z) = (x _y)^z:
Note that the dual of condition (1), that is,
(10) xz=)x^(y_z) = (x ^y)_z;
coincides with (1) due to commutativity. In other words, axiom (1) is
self-dual, therefore the principle of duality holds for modular lattices.
Condition (1) is not an identity, but it is easy to nd identities equivalent
to (1) within the class of lattices, thus showing that the class of modular
lattices is a subvariety of the class of all lattices. The most natural such
identity is perhaps
(2) x_(y^(x_z)) = (x _y)^(x_z):
39
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
40 2. Modular Lattices
For (1) implies (2) by taking z:=x_z, while (2) implies
xz=)x_(y^z) =x_(y^(x_z)) = (x _y)^(x_z) = (x _y)^z:
In view of the principle of duality, axiom (2) is also equivalent to its dual
(20) x^(y_(x^z)) = (x ^y)_(x^z):
Since the class of modular lattices is self-dual, it is natural to ask
whether it can also be de ned by a single self-dual identity. Such an identity
is well known. It is sometimes called the shearing identity:
(3) (x^(y_z))_(y^z) = (x _(y^z))^(y_z):
It is because of the name \shearing" that in subsequent pages we use the
notations(x;y;z ) ( rst occurrence in identity (16)). Clearly (1) implies (3)
by takingx:=y^z; y :=x; z :=y_z, while (3) implies
yz=)y_(x^z) = (y ^z)_(x^(y_z)) = (x _(y^z))^(y_z) = (y _x)^z :
Therefore, by adding the self-dual axiom (3) to any self-dual system of
axioms de ning lattices one obtains a self-dual system of axioms for modular
lattices. See also the end of x2, where identity (3) appears in the form (2.17).
We recall that a sublattice of a lattice (L; _;^) is a subset SLsuch
that for any x;y2Sit follows that x_y ; x ^y2S. The following
well-known characterization of modular lattices is very useful:
Theorem 2.1.1. A lattice is modular if and only if it does not include a
sublattice of the form in Fig.1.
rorxrzru
ry
SSA
A
A
Fig.1
Comment Fig.1 is a Hasse diagram, in which a segment with ends aand
b,abelowb, indicates that b covers a, that is, the following two conditions
are ful lled: a < b (meaningabanda6=b) and there is no element x
such thata<x<b. The lattice in Fig.1 is known as N5.
Proof: A lattice which includes a sublattice of the form in Fig.1 is not
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2.1 Modular Lattices within Lattices 41
modular, because x < z butx_(y^z) =x_o=x, while (x _y)^z=
u^z=z.
Conversely, suppose there exist x;y;z such thatxzbutx_(y^z)6=
(x_y)^z. Sincex_(y^x) =x= (x_y)^x, it follows that x6=z, hence
x<z . Sincex_(y^z)(x_y)^z, it follows that x_(y^z)<(x_y)^z.
Hencexywould imply y^z<y^z, a contradiction, while yxwould
imply the contradiction x < x ^z. Therefore xandyare incomparable.
Settingx^y=o, it follows that o < x ando < y . Similarly, setting
z_y=u, it follows that z < u andy < u. Thus we have obtained the
sublattice in Fig.1. 2
Corollary 2.1.1. (Kurosh [1935], Ore [1935]) A lattice is modular if and
only if
(4) xy&9z(x^z=y^z&x_z=y_z) =)x=y:
Proof: If the lattice is modular, the left-hand side of (4) implies
x=x_(x^z) =x_(z^y) = (x _z)^y= (y_z)^y=y:
If the lattice is not modular, the sublattice in Fig.1 shows that condition
(4) fails for y:=zandz:=y. 2
Corollary 2.1.2. A lattice is modular if and only if it satis es the identity
(5) x_(y^(x_z)) =x_(z^(x_y)):
Proof: If the lattice is modular, then from xx_zandxx_ywe
infer
x_(y^(x_z)) = (x _y)^(x_z) = (x _z)^(x_y) =x_(z^(x_y));
while if the lattice is not modular then (5) fails for the sublattice in Fig.1:
x_(y^(x_z)) =x_(y^z) =x_o=xandx_(z^(x_y)) =x_z=z:
2
Remark 2.1.1. Nishig ori [1954] has translated the modularity condition
in terms of segments.
Theorem 2.1.2. There is no system of several mutually independent iden-
tities characterizing modular lattices within the class of all lattices.
Proof: Suppose would be such a system. Then some member of
fails in the lattice in Fig.1, say f=g. Hencef=galready characterizes
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
42 2. Modular Lattices
modular lattices, therefore the existence of other members of contradicts
its independence. 2
2.2. De ning Modular Lattices in Terms of the Operations
_and^or by Other Tools
The rst direct de nition of modular lattices, that is, not including a system
of axioms for lattices, is due to Kolibiar [1956b], namely the independent
system
M1=fM1;M2g;
where
M1 ((x^y)^z)_(x^t) = ((t ^x)_(z^y))^x;
M2 (x_(y^y))^y=y:
Then Rie can [1957] devised the system
M2=fM3;M4;M5g;
where
M3 (x_y)^y=y;
M4 (x_y)_z=x_(y_z);
M5 (x^y)_(x^z) = ((z ^x)_y)^x;
and the independent system
M3=fM5;M6g;
where
M6 (x_(y_z))^z=z:
Soboci nski [1975a], being not aware of systems M2;M3, obtained the in-
dependent system
M4=fM7;M8g;
where
M7 (x^y)_(x^z) = ((z ^x)_(y_y))^x;
M8 (z_(y_x))^x=x;
then he discovered Rie can's paper and obtained another system M0
2by
replacing M 4by its dual in M2; cf. Soboci nski [1976a]. He also conjectured
that in M3axiom M 6cannot be replaced by (y _z)^z=z, which was
con rmed by Sudkamp [1976]. Quite independently, Ponticopoulos [1964]
suggested a 3-axiom system with 12 variables.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2.2 De ning Modular Lattices in Terms of Operations _,^or by Other Tools 43
Theorem 2.2.1. M 3is an independent system which de nes modular lat-
tices.
Proof: In every lattice condition M 6is ful lled, while M 5is equivalent
to the modularity condition (20). Conversely, suppose an algebra (L; _;^)
satis es M3. In view of the above remark, if we prove that Lis a lattice,
it will follow that it is a modular one.
By applying M 5withy:=y_x, then M 6withx:=z^x; z :=x, we
obtain
(1) (x^(y_x))_(x^z) = ((z ^x)_(y_x))^x=x:
By applying (1), then M 6withx:=y; y :=x^(y_x); z:=x^z, we infer
(2) (y_x)^(x^z) = (y _((x^(y_x))_(x^z)))^(x^z) =x^z :
By using in turn (1), then M 5withx:=y_x; y :=x^z; z:=x, and (2),
we get
x^(y_x) = ((x ^(y_x))_(x^z))^(y_x)
= ((y_x)^(x^z))_((y_x)^x) = (x ^z)_((y_x)^x) ;
we have thus proved that
(3) (x^z)_((y_x)^x) =x^(y_x):
Further we apply (1) and (3), both with y:=x; z :=x_x, which yields
x= (x^(x_x))_(x^(x_x)) = (x ^(x_x))_((x^(x_x))_((x_x)^x));
that is,x=y_(y_z), where we have set y=x^(x_x) andz= (x_x)^x.
By applying M 6we getx^z= (y_(y_z))^z=z, that is,
(4) x^((x_x)^x) = (x _x)^x:
The next computation uses in turn (3), (4), M 5withy:=x_x; z :=
(x_x)^xand M 6withx:= ((x _x)^x)^x; y :=z:=x:
x^(x_x) = (x ^(x_x))_((x_x)^x) = (x ^(x_x))_(x^((x_x)^x))
= ((((x _x)^x)^x)_(x_x))^x=x;
we have thus proved that
(5) x^(x_x) =x:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
44 2. Modular Lattices
A new application of (1) yields
(x^(x_x))_(x^(x_x)) =x
and taking into account (5) we obtain
(6) x_x=x:
It follows from M 6and (6) that
(7) (x_y)^y=y;
then (6) and (7) yield
(8) y^y=y:
Properties (8), M 5and (7) imply
(9) x_(x^z) = (x ^x)_(x^z) = ((z ^x)_x)^x=x:
From (7) and (9) we infer
(10) (x_y)_y= (x_y)_((x_y)^y) =x_y:
Further we apply in turn (8), (10), (8), M 5withx:=z:=x_y, then (7)
and (8):
x_y= (x_y)^(x_y) = ((x _y)_y)^(x_y)
= (((x _y)^(x_y))_y)^(x_y) = ((x _y)^y)_((x_y)^(x_y)) =y_(x_y);
hence
(11) y_(x_y) =x_y:
From (9), (11) and (9) we obtain
(12) (x^y)_x= (x^y)_(x_(x^y)) =x_(x^y) =x;
while from (6), M 5and (12) we get
x^y= (x^y)_(x^y) = ((y ^x)_y)^x=y^x;
that is,
(13) x^y=y^x:
It follows from (8), M 5, (8) and (12) that
(14) (x_y)^x= ((x ^x)_y)^x= (x^y)_(x^x) = (x ^y)_x=x:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2.2 De ning Modular Lattices in Terms of Operations _,^or by Other Tools 45
Now we apply (14), (7), M 5, (13) and (7):
x_y= ((x _y)^x)_((x_y)^y) = ((y ^(x_y))_x)^(x_y)
= (((x _y)^y)_x)^(x_y) = (y _x)^(x_y);
hence
x_y= (y_x)^(x_y);
therefore
y_x= (x_y)^(y_x);
and taking into account (13) it follows that
(15) x_y=y_x:
Summarizing, we have proved that M 5and M 6imply the commutativity
of the two operations, i.e., (15), (13), their idempotency (6), (8) and the
two absorption laws (12), (7). We are going to use freely these properties
in proving the associativity of the two operations.
Set
x_(y_z) =a
and note that a^x= ((y _z)_x)^x=xby (7) and a^y=yby M 6,
hence M 5implies
x_y= (a^x)_(a^y) =a^(x_(y^a)) =a^(x_y);
therefore, taking into account that a^z=zby M 6and using M 5, we have
(x_y)_z= (a^(x_y))_(a^z)
=a^((x_y)_(z^a)) =a^((x_y)_z):
Thus
(x_y)_z= (x_(y_z))^((x_y)_z);
which implies
x_(y_z) = (z _y)_x= (z_(y_x))^((z_y)_x)
= (x_(y_z))^((x_y)_z) = (x _y)_z :
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
46 2. Modular Lattices
We have thus proved that M 5and M 6imply the associativity M 4. There-
fore, in order to prove the dual of M 4it suces to show that the duals of
M5and M 6are also valid. But
(x_y)^(x_z) = ((x ^(x_z))_y)^(x_z)
= ((x _z)^y)_((x_z)^x) =x_(y^(z_x))
by M 5withx:=x_z; z:=x, while M 4implies
(x^(y^z))_z= (x^(y^z))_((y^z)_z)
= ((x ^(y^z))_(y^z))_z= (y^z)_z=z:
Finally consider a set fa;bg. The operations x_y=x^y=aful l M 5
but not M 6, while the operations x_y=x^y=ysatisfy M 6but not M 5.
2
A somewhat surprising independent system was given by Vaida [1957],
namely
M5=fM9;M10g;
where
M9x^((x_y)_z) =x
and M 10is an ingenious translation of (1):
M10less(x;z ) =)z^(y_x) =x_(y^z);
where we have used the notation
less(x;y )()x=zor9z0x=z0^zor9z0x=z^z0or9z0z=z0_x:
Kolibiar [1956b] characterized the class of modular lattices with 1 by
the system
M6=fM11;M12;M13g;
where
M11 ((x^y)^z)_(x^t) = ((t ^x)_(z^y))^x;
M12x_1 = 1;
M13 1^x=x;
while Tamura [1975] improved this result by showing that the system
M7=fM14;M15g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2.2 De ning Modular Lattices in Terms of Operations _,^or by Other Tools 47
where
M14 (x_(y^y))^y=y;
M15 (((0_x)^y)^z)_(x^t) = ((t ^x)_(z^y))^x;
de nes modular lattices with 0. Soboci nski [1979] de ned modular lattices
with 0 and 1 by a system with two identities again, having 8 and 3 variables,
respectively.
The relation of lattice betweenness
(1.4.12) axb() (a^x)_(x^b) =x= (a_x)^(x_b);
already introduced in Chapter 1, was also used for de ning modular lattices.
Smiley and Transue [1943] proved the following
Theorem 2.2.2. Let(L;_;^;0)be an algebra of type (2,2,0) endowed with
a ternary relation (4.1.12) satisfying L44;L45;L46and
M16abc&adb=)adc:
Then: )L is a modular lattice with least element 0 and order relation
ab() 0abora=b
if and only if it satis es L48andL47orL48and
M17 0bc& 0bd &cxd=)0bx:
)When the foregoing holds, axb is the lattice betweenness (1.4.12).
L.M. Kelly [1952] extended the above theorem to arbitrary modular
lattices, while Kolibiar [1958] characterized modular lattices in terms of
K-segments.
Setting
(16) s(x;y;z ) = ((y ^z)_x)^(y_z);
the characterization (1.3) of modularity within Lcan be written in the form
(17) s(x;y;z ) = ~s(x;y;z );
where ~ is the operation of taking the dual. Several variations on this
theme have been elaborated in the literature, as we are going to show in
the remainder of this chapter.
Theorem 2.2.3. (Hashimoto [1951]) In a bounded modular lattice
(L;_;^;0;1)the ternary operation (16)satis es (17)and
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
48 2. Modular Lattices
M18s(0;x; 1) =x;
M19s(x;y;y ) =y;
M20s(s(x;v;w );y;s(z;v;w )) =s(x;s(y;w;v );s(z;w;v ));
(18) x_y=s(x;1;y); x^y=s(x;0;y):
Conversely, if an algebra (L;s; 0;1)of type (3,0,0) satis es M18 M20,
then (L;_;^;0;1), where _and^are de ned by (18), is a bounded modular
lattice in which (16)and(17)hold.
Kolibiar and Marcisov a [1974] proved a similar theorem, in which axiom
M20is replaced by the axioms
M21s(x;y;z ) =s(x;z;y );
M22s(s(x;v;z );y;z ) =s(x;s(y;v;z );z):
We have seen in Chapter 1, x4, the characterization of lattices given by
Martin [1965] in terms of the operation t(x;y;z ) = (x _z)^(y_(x^z));
note thatt(x;y;z ) =s(y;x;z ) due to the commutativity of _and^. In
the same paper, Martin characterized modular lattices by axioms L 55, L58,
L59and
M23t(x;z;x) = x;
M24t(t(x; 0;z);1;t(y; 0;t(x; 1;z)) =t(x;y;z );
M25t(t(x; 1;z);0;t(y; 1;t(x; 0;z)) =t(x;y;z ):
It follows from Theorem 3 in D. Kelly and Padmanabhan [1989] that
a ternary lattice term pyields the join if and only if pt_(y^z) for a
permutation (t;y;z ) of its variables. Therefore no ternary term pcan yield
both the join and the meet, because the above inequality together either
withpt^(y_z) or withpy^(t_z), yieldty_z, which need not
hold.
So, while a 4-ary lattice term, e.g. (x ^y)_(z^u), can yield both _
and^, a ternary lattice term can do this only with the aid of the constants
0 and 1.
The following result is also relevant:
Theorem 2.2.4. (Padmanabhan and Penner [2004]) Let L be a subdirectly
irreducible lattice. If p(x;y;z )is an essentially ternary term such that
p(x;c;y )is a semilattice operation on L, then either L is bounded above
withc= 1andp(x;1;y) =x_y, or L is bounded below with c= 0and
p(x;0;y) =x^y.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2.3 Self-Dual Equational Bases for Modular Varieties 49
2.3. Self-Dual Equational Bases for Modular Varieties
In this section we consider a non-trivial nitely-based equational class Kof
modular lattices. In view of Corollary 1.2.3, Kcan be characterized by two
identities and according to Theorem 1.3.3, Kis not one-based. However
the number n+ 5 of variables occurring in axiom L 28of Corollary 1.2.3 can
be improved: we provide below a 3-basis involving n+ 3 variables. Then
we explore the possibility of obtaining self-dual equational bases of K; in
particular we obtain a single-identity characterization relative to the class
Mof modular lattices.
To state our rst result we use Theorem 1.2.3 and suppose, without loss
of generality, that Kis de ned by a single identity f=galong with the
axioms of modular lattices. Let us introduce the following axiom:
(1) ((x^f)^z)_(x^t) = ((t ^x)_(z^g))^x;
where the variables x;z;t do not occur in forg.
Theorem 2.3.1. The following system de nes K:(1)and
L_
7 (x^y)_y=y;
L^
7 (x_y)^y=y:
Proof: IfL2Kthen L_
7and L^
7are ful lled and taking y:=f=gin M 1
we obtain (1). Conversely, suppose Lsatis es (1), L_
7and L^
7. According
to a well-known argument (cf. L1in Ch.1,×2), the two absorption laws
L_
7and L^
7imply idempotency: y_y=yandy^y=y, therefore L^
7
implies M 2. Besides, denoting the variables of fandgbyy1;:::;y nand
takingy1:=:=yn:=y, we obtain f(y1;:::;y n) =y=g(y1;:::;y n)
by Corollary 1.2.1, hence (1) reduces to M 1. So Kolibiar's system M1is
ful lled, showing that Lis a modular lattice. Now set
(2) y1_:::_yn=u; y 1^:::^yn=o
and note that any lattice polynomial psatis es
(3) op(y1;:::;y n)u:
the easy proof is by induction on the de nition of polynomials. Therefore,
takingx:=z:=uandt:=oin (1) we obtain f=g, henceL2K. 2
In the remainder of this section we are looking for self-dual equational
bases which de ne modular varieties relative to the class Mor to the class
Lof all lattices, using the function
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
50 2. Modular Lattices
(2.16) s(x;y;z ) = (x _(y^z))^(y_z);
and the characterization (2.17) s= ~sof modularity within L.
The following lattice identities will be needed:
(4) s(x;y ^z;y_z) =s(x;y;z );~s(x;y_z;y^z) = ~s(x;y;z ):
Theorem 2.3.2. (D. Kelly and Padmanabhan [2002]) Ifp=qand~p= ~q
de ne a variety Krelative to M, then the self-dual identity
(5) s(x;y ^p;y_~q) = ~s(x;y _~p;y^q);
where the variables x,y do not occur in p or q, de nes Krelative to L.
Proof: IfL2Kthen (5) is readily veri ed. Conversely, suppose a lattice
Lsatis es (5), that is,
(x_((y ^p)^(y _~q)))^((y ^p)_(y _~q)) = (x^((y _~p)_(y^q)))_((y _~p)^(y^q)):
Takex:=o; y:=u, cf. (2), where y1;:::;y nare the variables of pandq. It
follows by (3) that p=qinLand since (5) is self-dual, it also implies that
Lsatis es ~p= ~q. Besides, taking y1:=:=yn:=z, identity (5) becomes
s(x;y ^z;y_z) = ~s(x;y _z;y^z). Therefore, taking into account (4),
we obtain (2.17), showing that L2M. Summarizing, we have obtained
L2K. 2
Theorem 2.3.3. (D. Kelly and Padmanabhan [2002]) Ifp=qand~p= ~q
de ne a variety Krelative to M, andpqholds in L, then the self-dual
identity
(6) s(x;p; ~q) = ~s(x;~p;q);
where the variable x does not occur in p or q, de nes Krelative to M.
Moreover, if p=qimplies modularity, then (6)de nes Krelative to L.
Proof: IfL2KthenL2M, hencep=qand ~p= ~q. Using the
symmetry of s(x;y;z ) iny;zand the characterization (2.17) of modularity,
we obtain (6):
s(x;p; ~q) =s(x;q; ~p) =s(x;~p;q) = ~s(x;~p;q):
Conversely, suppose a modular lattice Lsatis es (6). By the same
technique as in the proofs of Theorems 3.1 and 3.2, we nd elements o;u2L
such thatop;~p; q; ~qu. Then from pqwe deduce ~q~pand
p^~qp^~pq^~p=s(o;q; ~p) = ~s(o;~q;p) =p^~q;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
2.3 Self-Dual Equational Bases for Modular Varieties 51
thereforep^~q=p^~p=q^~p. We infer similarly p_~q=p_~p=q_~p, by
usinguinstead ofo. Fromp^~p=q^~pandp_~p=q_~pwe obtainp=q
by Corollary 1.1. Interchanging the identities p=qand ~p= ~qwill give a
proof that ~ p= ~qholds inL, thereby L2K, completing the proof of the
rst statement.
Now assume that p=qimplies modularity and let Lbe a lattice sat-
isfying (6). If we show that (6) implies modularity, it will follow from the
rst part of the theorem that L 2K. So let us prove that (6) fails in any
non-modular lattice. In view of Theorem 1.1, it suces to prove this for
the ve-element lattice N5 depicted in Fig.1, but in which we change the
notation:x=a; y =b; z=c.
The identity p=qfails in the non-modular lattice N5 and since pq
holds in any lattice, it follows that p<q for a suitable substitution of the
variables. Several cases are possible.
1. Ifp=othen ~p=ufor the same substitution instance (take the
upside down version of N5). Hence condition (6), which we write explicitly,
(60) (x_(p^~q))^(p_~q) = (x ^(~p_q))_(~p^q);
reduces tox^~q=x_q. Takingx:=owe see that this equality fails.
2. Ifp=athen ~p=cand there are two subcases.
2.1. Ifq=cthen ~q=aand (60) reduces to (x _a)^a= (x^c)_c,
which is false.
2.2. Ifq=uthen ~q=oand (60) reduces to x^a=x_a, which is false.
3. Ifp=cthen ~p=aandq=u, hence ~q=o, therefore (60) reduces to
x^c=x_a, which is false.
4. Ifp=bthen ~p=bandq=u, hence ~q=o, therefore (60) reduces to
x^b=x_b, which is false.
Thus condition (6) fails in all cases. 2
Corollary 2.3.1. (D. Kelly and Padmanabhan [2002]) Every nitely-based
self-dual variety of modular lattices can be de ned by a single self-dual iden-
tity modulo the lattice axioms.
Proof: Apply Theorem 1.2.3 to the identities de ning the variety Kand
to the characterization ~ s=sof modularity; cf.(2.17). Then the resulting
equationp=qcharacterizes K,p=qimplies modularity, and since ~ ss
holds in L, the proof of Theorem 1.2.3 immediately shows that pqholds
inL. Since Kis self-dual, ~ p= ~qalso holds in K, so that p=q& ~p= ~qis
a (redundant) de nition of K. Now the desired result follows by Theorem
3.3. 2
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
52 2. Modular Lattices
Finally let us con ne to the variety of all modular lattices. We have seen
in Chapter 1,×3 that the class of all lattices is characterized by the self-dual
system L8consisting of (possibly variants of) the two absorption laws and
two \skew associativity" laws (x y)z= (yz)xfor=_;^. We have
also noted in this chapter that identity (1.3) is equivalent to modularity.
Therefore the self-dual system
M8=L8[ f(1:3) g
characterizes modular lattices. Let us prove it is independent.
Any non-modular lattices proves the independence of (1.3). The other
independence models will be based on the set f0;1g. Taking x_y= 1 and
x^y= min(x;y ) proves the independence of L_
7. Takingx_y=yand
x^y= min(x;y ) proves the independence of L_
33. The dual models prove
the independence of L^
7and L^
33, respectively.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Chapter 3
Distributive Lattices
The basic properties of modular and distributive lattices were discovered by
Dedekind [1897] and Schr oder [1890-1905]; cf. Birkho [1948]. Any chain
is a distributive lattice; the open subsets of every topological space form
a distributive lattice and so do the closed subsets. As a matter of fact,
there are many examples of distributive lattices, including the important
subclass of Boolean algebras; cf. next chapter.
We begin this chapter with some of the most useful characterizations of
distributive lattices within the class of all lattices. Then we survey direct
de nitions of distributive lattices in terms of disjunction and conjunction
and for bounded distributive lattices in terms of a ternary operation.
3.1. Distributive Lattices within Lattices
Distributivity is a natural condition for lattices, very much like commuta-
tivity for groups. As a matter of fact, in the early days of lattice theory,
towards the end of XIX century, it was believed that all lattices were dis-
tributive. It was Schr oder [1890-1905] who rst proved the equivalence of
conditions D 1below and noted that calculus with groups is not distributive
(cf. Anhangen 4,6). Curiously enough, Schr oder could not convince Peirce;
cf. Birkho [1948].
A lattice (L; _;^) is said to be distributive provided
D^
1x^(y_z) = (x ^y)_(x^z):
It is immediately seen that condition D^
1implies the identity (2.1.10) de n-
ing modular lattices, therefore any distributive lattice is modular. The con-
verse does not hold, as shown e.g. by the lattice in Fig.2 below, which is
modular by Theorem 2.1.1, but not distributive.
An important property is that in any lattice condition D^
1is equivalent
to its dual
D_
1x_(y^z) = (x _y)^(x_z):
53
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
54 3. Distributive Lattices
For it follows from D^
1that
(x_y)^(x_z) = ((x _y)^x)_((x_y)^z)
=x_((x^z)_(y^z)) = (x _(x^z))_(y^z) =x_(y^z):
So D^
1=)D_
1, hence D_
1=)D^
1by duality. Therefore the principle of
duality holds for distributive lattices.
Each of the following dual inequalities characterizes distributive lattices
because the converse inequalities hold in every lattice:
(1) (x_y)^(x_z)x_(y^z);
(10) x^(y_z)(x^y)_(x^z):
Lemma 3.1.1. If a lattice is modular but not distributive then it includes
a sublattice of the form in Fig.2.
rr r rr
ox y zu
@@@
@@@
Fig.2
Proof: Suppose the elements x;y;z fail to satisfy D_
1. Since
ac=)a_(b^c) = (a _b)^c= (a_b)^(a_c);
it follows that x6z. Since
c<a =)a_(b^c) =a=a^(a_b) = (a _c)^(a_b);
it follows that z6<x. Therefore xandzare incomparable, and similarly so
arexandy. Since
bc=)a_(b^c) =a_b= (a_b)^(a_c);
it follows that y6zand similarly z6y. Therefore yandzare incompa-
rable.
Summarizing, the elements x;y;z are pairwise incomparable. Then x^
y<y<y _zand setting o=x^y^zwe geto<z <y _z, becauseo=z
would imply the contradiction zx^y. Assuming o<x^y, the elements
o;x^y;y;z;y _zwould be related as shown in Fig.1, which contradicts
the modularity. Therefore o=x^yand since this conclusion has been
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.1 Distributive Lattices within Lattices 55
obtained only from the pairwise incomparability of x;y;z , it follows by
symmetry that o=y^zando=x^z. We deduce analogously that
x_y=y_z=x_z=u. Therefore the elements o;x;y;z;u are related as
shown in Fig.2. 2
Theorem 3.1.1. A lattice is distributive if and only if it includes neither
a sublattice of the form in Fig.1 (Chapter 2), nor a sublattice of the form
inFig.2.
Comment Loosely speaking, Theorems 2.1.1 and 1.1 characterize a struc-
ture by \forbidding" certain substructures. This reminds a famous theorem
in graph theory, which characterizes the planarity of a graph by forbidding
certain subgraphs. However we do not know other theorems of this type in
algebra.
Proof: If a lattice is distributive, then it is modular, hence by Theorem
2.1.1 it does not include a sublattice of the form in Fig.1, and it does not
include a sublattice of the form in Fig.2 either, because the latter is not
distributive.
Conversely, suppose the lattice is not distributive. If it is not modular,
then by Theorem 2.1.1 it includes a sublattice as depicted in Fig.1, while
in the opposite case it includes a lattice as in Fig.2 by Lemma 1.1. 2
Remark 3.1.1. The lattice in Fig.1 is the ve-element non-modular lat-
tice; let us call it N5. The modular non-distributive lattice in Fig.2 has 3
pairwise incomparable elements; let us call it M3. Then Theorem 1.1 can
be rephrased as follows: A lattice is distributive if and only if it includes
neither N5, nor M3.
We can characterize the self dual variety of distributive lattices by a self-
dual identity:
Corollary 3.1.1. A lattice is distributive if and only if it satis es
(2) (x^y)_(y^z)_(x^z) = (x _y)^(y_z)^(x_z):
Proof: If the lattice is distributive, then
(x^y )_(y^z)_(x^z ) = ((x_z )^y)_(x^z ) = (((x_z )^y)_x)^(((x_z )^y)_z)
= (x_z_x)^(y_x)^(x_z_z)^(y_z) = (x _z)^(x_y)^(y_z):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
56 3. Distributive Lattices
If the lattice is not distributive then it includes at least one of the
sublattices N5, M3. But in N5
(x^y)_(y^z)_(x^z) =o_o_x=x6= (x_y)^(y_z)^(x_z) =u^u^z=z;
while in M3
(x^y)_(y^z)_(x^z) =o6= (x_y)^(y_z)^(x_z) =u:
2
Corollary 3.1.2. A lattice is distributive if and only if it satis es
(3) (x_y)^(z_(x^y)) = (x ^y)_(y^z)_(z^x):
Proof: Identity (3) is immediate in a distributive lattice and it fails in a
non-distributive lattice because in N5 and M3 we have (x _y)^(z_(x^y)) =
u^z=z, while in N5 we have (x ^y)_(y^z)_(z^x) =o_x=x, while
in M3 we have (x ^y)_(y^z)_(z^x) =o. 2
The following analogue of Corollary 2.1.1 is valid:
Corollary 3.1.3. A lattice is distributive if and only if
(4) 9z(x^z=y^z&x_z=y_z) =)x=y:
Proof: If the lattice is distributive, the left-hand side of (4) implies
x=x^(x_z) =x^(y_z) = (x ^y)_(x^z)
= (x^y)_(y^z) = (x _z)^y= (y_z)^y=y:
If the lattice is not distributive, then it includes at least one of the
sublattices N5, M3. In N5 we have x^y=o=z^yandx_y=u=z_y
butx6=z, while in M3 we have x^z=o=y^zandx_z=u=y_z
butx6=y. 2
Several modi cations of condition (4) are equivalent to it:
Corollary 3.1.4. Each of the following conditions is equivalent to the dis-
tributivity of the lattice:
(5) 9z(x^zy^z&x_zy_z) =)xy;
(6) 9z(x^zy&xy_z) =)xy;
(7) 9z(x^zy&xy_z) =) 8z (x^zy&xy_z):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.1 Distributive Lattices within Lattices 57
Proof: (4)=)(5): The left-hand side of (5) implies
x^z=x^z^y^z=x^y^z ;
x_z= (x_z)^(y_z) = (x ^y)_z ;
thereforex=x^y, that is,xy.
(5)=)(4) because a=b()ab&ba.
(5)()(6) because their left-hand sides are clearly equivalent.
(6)=)(7) because xyimplies the right-hand side of (7).
(7)=)(6): take z:=xin the right-hand side of (7). 2
Conditions (6) and (7) are due to Picu [1982], who provided more vari-
ations on this theme.
Theorem 1.1 suggests the question of whether one can de ne distributive
lattices satisfying two mutually independent identities: one identity should
\destroy" N5 and the other would do the same job for M3. Here is such an
equational basis:
Theorem 3.1.2. A lattice is distributive if and only if it satis es the mu-
tually independent axioms
(8) x_(y^(x_z)) = (x _y)^(x_z);
(9) x^(y_z) =x^(y_(x^(z_(x^y)))):
Proof: Note that (8) is the modularity law (2.1.2). In a distributive
lattice the right-hand side of (9) equals
x^(y_(x^z)_(x^y)) = (x ^y)_(x^z) =x^(y_z):
Conversely, a lattice which satis es (8) and (9) is distributive since it does
not include N5 because of (8), while M3 is also forbidden because in M3 we
havex^(y_z) =xbut
x^(y_(x^(z_(x^y)))) =x^(y_(x^z)) =x^y=o:
The axioms are independent because M3 satris es (8), while N5 satis es
(9):
x^(y_(x^(z_(x^y)))) =x^(y_(x^z)) =x^(y_x) =x=x^(y_z):
2
From the axiomatic point of view, this result is interesting because it
is impossible to characterize distributivity (modulo lattice theory) by any
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
58 3. Distributive Lattices
independent set with more than two identities. This is simply an equational
reformulation of Theorem 1.1. Similarly, modularity cannot be character-
ized by any independent system with more than one identity!
Theorem 3.1.3. Ifis a system of mutually independent identities char-
acterizing distributive lattices within the class of all lattices, then has at
most two axioms.
Proof: Some member of fails in N5, say f1=g1. If this axiom also
fails in M3, then it already characterizes distributive lattices and since
is independent, it follows that = ff1=g1g. Otherwise some other
member of fails in M3, say f2=g2. Then the system ff1=g1;f2=g2g
characterizes distributive lattices and since is independent, it reduces to
this system. 2
By sharp contrast, Abelian groups can be characterized modulo group
theory by a set of nindependent identities, for all n. In other words, the
ability to characterize an equational class by demanding the non-occurrence
of certain lattices is a unique feature of lattice theory and has no apparent
counter-part in other algebras like groups or rings.
3.2. De ning Distributive Lattices in Terms of the Opera-
tions _and^
The rst direct de nition of distributive lattices, that is, not including a
system of axioms for lattices, is the system
D1=fL^
1;L_
2;L^
2;L^
3;L^
4;D^
1g;
suggested by Birkho [1948], Ch.IX,x1,Exercise 6 and rediscovered by
Felscher [1958]. The proof is very easy: by de ning xy()x^y=x,
we obtain a meet semilattice (L; ^). To prove that (L; _;^) is a lattice, we
note thatxx_yby L^
4,yx_yby L_
2and L^
4, whilexzandyz
implyx_yzbecause
(x_y)^z=z^(x_y) = (z ^x)_(z^y) = (x ^z)_(y^z) =x_y:
Ponticopoulos [1962] provided two 3-identity bases involving 12 variables
each. The 9-axiom system devised by Ellis [1949] uses D 1, L2, L3, a variant
of L1and (1.2.1).
Two other systems were given by Rudeanu and Vaida [2004]:
D2=fL^
1;L_
2;L^
2;L_
4;D^
1g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.2 De ning Distributive Lattices in Terms of the Operations _and^ 59
D3=fL^
1;L_
4;L_
5;L_
6;L_
7;D^
1g:
In view of McKenzie's Theorem 1.3.3, the variety of distributive lattices
cannot be de ned by a single identity. The following two-identity base has
been widely used in the literature:
Theorem 3.2.1. (Sholander [1951]) An algebra (L;_;^)of type (2,2)is a
distributive lattice if and only if it satis es the system
D4=fL^
4;D^
2g;
where
L^
4x^(x_y) =x;
D^
2x^(y_z) = (z ^x)_(y^x);
Comment See Appendix A for a proof provided by the computer program
Prover9.
Proof: Necessity is trivial. Conversely, suppose Lsatis es D4. We rst
prove the idempotency laws and the commutativity of ^. The hypothesis
implies
(1) x=x^(x_x) = (x ^x)_(x^x) ;
then from (1) and L^
4we obtain
(2) x^x= (x^x)^((x^x)_(x^x)) = (x ^x)^x
and using (1), D^
2, (2) and (1) we deduce
x^x=x^((x^x)_(x^x)) = ((x ^x)^x)_((x^x)^x) = (x ^x)_(x^x) =x;
so that (1) becomes x=x_x. This yields further, via D^
2,
x^y= (x^y)_(x^y) =y^(x_x) =y^x:
Now we use freely idempotency, the commutativity of ^andD4to
obtain the commutativity of _. We have in turn
(3) x=x^(x_y) = (y ^x)_(x^x) = (y ^x)_x;
(4)x=x^x=x^((y^x)_x)
= (x^x)_((y^x)^x) =x_((y^x)^x);
(5)x_y= (x_y)^(x_y) = (y ^(x_y))_(x^(x_y))
= ((y^y)_(x^y))_x= (y_(x^y))_x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
60 3. Distributive Lattices
and it follows from (3), (4) and (5) that
(6) y= (x^y)_y= (y_((x^y)^y))_(x^y) =y_(x^y) ;
from (4) and (6) we get x_y=y_x.
We have thus established properties L 1;L2and L 4. We prepare the
proof of L_
3by computing
c^((a_b)_c) = (c ^(a_b))_(c^c) = (c ^(a_b))_c=c;
b^((a_b)_c) = (b ^(a_b))_(b^c) =b_(b^c) =b;
and similarly a^((a_b)_c) =a. Set (a _b)_c=Panda_(b_c) =Q.
Then, using D^
2, we obtain
Q= (a^P)_((b^P)_(c^P)) = (a ^P)_((b_c)^P) = (a _(b_c))^P=Q^P
and quite similarly we deduce P=P^Q, therefore P=Q, that is, L_
3.
In the above proof of L_
3we have used the distributivity D^
2, which
reduces to D^
1. Note that the proof of the implication D^
1=)D_
1, given
in the beginning of this chapter, uses in fact D^
1;L2, L4and L_
3, therefore
D4implies also D_
1. It follows that the proof of L_
3can be dualized and we
obtain L^
3as well. 2
We now pass to direct de nitions of distributive lattices with 0 or with
1 or with 0 and 1, that is, we include 0 and/or 1 in the type of the algebra,
even in the cases when the original papers work with algebras of type (2,2);
see e.g. the next system.
The rst such system of axioms, namely
D1
1=fL^
3;D^
1;D^
3;L_1;L1_;L^1;L1^g;
where we have set
D^
3 (y_z)^x= (y^x)_(z^x);
L_1x_1 = 1; L1_ 1_x= 1;
L^1x^1 =x; L1^ 1^x=x;
de nes distributive lattices with greatest element 1 and is essentially due to
G.D.Birkho and G. Birkho [1946]; see also Birkho [1948], Ch.IX, Theo-
rem 3. The problem of the independence of this system, known as Birkho 's
Problem 65, was solved in the armative by Croisot [1951], Wooyenaka
[1951], Matusima [1952], Sz asz [1952] and Zelinka [1967], independently of
each other.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.2 De ning Distributive Lattices in Terms of the Operations _and^ 61
As a matter of fact, in the original paper axioms L _1;:::; L1^appear
in the form \there is 1 such that x_1 = 1 for all x" etc. However, the
authors implicitly admit that the element 1 is the same in all the axioms. As
explained above, we disregard this formulation and interpret D1
1as referring
to an algebra (L; _;^;0;1) of type (2,2,0,0); so did Matusima, Sz asz and
Zelinka. In his paper, Croisot puts the questionable axioms in the form
91x_1 = 1; 91010_x= 10;
9100x^10=x; 910001000^x=x;
adds the axiom 1 = 100, states that one can prove 1 = 10and 100= 1000, and
proves the independence of the new system. He then proves that axioms
10_x= 1;1000^x=x;D^
1andD^
3can be replaced by D^
2,yand this reduced
system is also independent. Wooyenaka provided four independent systems
by a similar approach.
Soboci nski [1972a,b] provided a variant of the Croisot system, then sim-
pli ed it as follows:
D1
2=fD^
4;L_1;L^1g;
where
D^
4x^((y^y)_z) = (z ^x)_(y^x):
The simplest direct de nition of distributive lattices with 1 is the dual
of a theorem which characterizes distributive lattices with 0 and was found
by Ferentinou-Nicolacopoulou [1969] and Tamura [1975], independently of
each other:
Theorem 3.2.2. Each of the following systems of axioms characterizes
distributive lattices with greatest element within algebras (L;_;^;1)of type
(2,2,0):
D1
3=fD_
5;L_
4g;
D1
4=fD_
6;L_
4g;
where
D_
5x_(y^z) = (z _(x^1))^(y_(x^1));
D_
6x_(y^z) = (z _(x^1))_(y^x):
Proof: We begin with D1
3. Necessity is trivial. Conversely, suppose L
satis es D1
2. Then we obtain in turn
x_(x^1) =x;
yThis axiom appears simultaneously in the papers by Croisot and Sholander.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
62 3. Distributive Lattices
x=x_(x^x) = (x _(x^1))^(x_(x^1)) =x^x;
x=x_(x^x) =x_x;
x_y=x_(y^y) = (y _(x^1))^(y_(x^1)) =y_(x^1);
and from the latter identity with y:=xthen withy:=x^1, we get
x=x_x=x_(x^1) = (x ^1)_(x^1) =x^1;
hence D_
5reduces to the dual D_
2of D^
2, therefore (L; _;^) is a distributive
lattice by the dual of Theorem 2.1 and it satis es x=x^1 as well.
System D1
4is left to the reader. 2
Another 2-axiom system for distributive lattices with 1 was given by
Tamura [1975].
Is eki and ^Ohashi [1970] presented (without proofs) four equational bases
for bounded distributive lattices, each of them having 5 axioms and involv-
ing 6,6,7 and 8 variables, respectively. One of these systems corrects an
error in ^Ohashi [1968], pointed out by Lowig [1969]. A two-axiom system
involving 8 variables was provided by Soboci nski [1979].
Note that a short de nition of bounded distributive lattices can be ob-
tained by adding one axiom to a short de nition of distributive lattices with
1/with 0, or else, according to a remark of Sholander [1951], by adding the
axiom 0 _(x^1) =xor its dual to a short de nition of distributive lat-
tices. For this axiom can be written in the form (0 _x)^(0_1) =x,
hencex0_1, showing that 0 _1 is the greatest element, so that the
latter identity reduces to 0 _x=x, therefore the original axiom becomes
x^1 =x.
3.3. De ning Bounded Distributive Lattices in Terms of the
Median Operation
Recall that a latice with 0 and 1 is said to be bounded. This section is de-
voted to various characterizations of bounded distributive lattices in terms
of the median operation
(1) m(x;y;z ) = (y ^z)_(x^z)_(x^y);
which was already given as an example of a majority polynomial in arbitrary
lattices.
It was seen in Corollary 1.1 that distributive lattices are characterized
by the identity m= ~m, so that we have also
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.3 De ning Bounded Distributive Lattices in Terms of the Median Operation 63
(2) m(x;y;z ) = (y _z)^(x_z)^(x_y):
As a matter of fact
m(x;y;z ) = (y ^z)_(x^(y_z)) = ~s(x;y;z );
wheresis the function studied in Chapter 2, where it was seen that modular
lattices are characterized by the identity s= ~s.
In a bounded distributive lattice identities (2) and (1) imply
(3) x_y=m(x; 1;y); x ^y=m(x; 0;y):
The results of this section fall under the following general scheme. A set
Pof polynomial identities of type (3,0,0) is said to characterize bounded
distributive lattices if the following hold: ) the median operation (1) of
every bounded distributive lattice (L; _;^;0;1) satis es P, and ) con-
versely, if an algebra (L;m; 0;1) of type (3,0,0) satis es P, then the algebra
(L;_;^;0;1) de ned by (3) is a bounded distributive lattice in which (1)
holds.
From a practical point of view, the above point ) is a matter of rou-
tine computation which raises no diculty, so that the proofs reduce to
establishing ).
Lemma 3.3.1. If a ternary operation satis es
(x;y;z ) = (y;x;z ) and (x;y;z ) = (x;z;y );
then is invariant under any permutation of the variables.
Proof: Follows from
(x;y;z ) = (y;x;z ) = (y;z;x) = (z;y;x) = (z;x;y ) = (x;z;y ):
2
Theorem 3.3.1. (Birkho and Kiss [1947]) Bounded distributive lattices
are characterized by the system
D01
1=fD6;D7;D8;D9;D10g;
where
D6m(x;y;z ) =m(y;x;z );
D7m(x;y;z ) =m(x;z;y );
D8m(x;y;x) = x;
D9m(0;x; 1) =x;
D10m(m(x;y;z );t;v ) =m(m(x;t;v );y;m(z;t;v )):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
64 3. Distributive Lattices
Proof: It follows from D 6and D 7via Lemma 3.1 that mis invariant
under any permutation of the variables, therefore the operations _and^
de ned by (3) are commutative, hence in order to check rst system D1
1it
suces to prove L^
3;L_1;L^1;D^
1:
x^x=m(x; 0;x) =x;
x_1 =m(x; 1;1) =m(1;x; 1) = 1;
x^1 =m(x; 0;1) =m(0;x; 1) =x;
x^(y_z) =m(x; 0;m(y; 1;z)) =m(m(y; 1;z);0;x)
=m(m(y; 0;x); 1;m(z; 0;x)) =m(x^y;1;x^z) = (x ^y)_(x^z);
x^0 =m(x; 0;0) =m(0;x; 0) = 0:
The last point is to prove identity (1). Indeed, by applying in turn (3),
(3), symmetry, D 10, symmetry, D 8and (3), we obtain
(x^y)^m(x;y;z ) =m(x^y;0;m(x;y;z )) =m(m(x; 0;y);0;m(x;y;z ))
=m(m(0;x;y );0;m(z;x;y )) =m(m(0; 0;z);x;y )
=m(m(0;z; 0);x;y ) =m(0;x;y ) =x^y;
hencex^ym(x;y;z ) and similarly y^zm(x;y;z ) andz^x
m(x;y;z ), therefore
(x^y)_(y^z)_(z^x)m(x;y;z )
and a similar proof shows that
m(x;y;z )(x_y)^(y_z)^(z_x);
whence identity (1) follows by Corollary 3.1.1. 2
The independence of the above system was not studied. Cheremisin
[1958] claimed that axiom D 7follows from the others, but this is wrong.
However Birkho [1948] suggested that one of the axioms D 6and D 7or
both can be dispensed with if axiom D 10is suitably modi ed; this is known
as Birkho 's Problem 64. Several solutiona have been proposed.
The rst solutions to Problem 64 were given by Vassiliou [1950]*, the
simplest one being
D01
2=fD8;D9;D11g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.3 De ning Bounded Distributive Lattices in Terms of the Median Operation 65
where
D11m(x;m(y;z;t);v ) =m(m(v;x;t);z;m(v;x;y )):
Several authors found other solutions: Croisot [1951], who gave the
independent system
D01
3=fD8;D9;D12g;
where
D12m(x;m(y;z;t);v ) =m(z;m(t;x;v );m(y;x;v ));
Hashimoto [1951], who devised the systems
D01
4=fD8;D9;D13g;
D01
5=fD8;D9;D14g;
where
D13m(x;m(y;z;t);v ) =m(m(v;z;x);y;m(v;t;x)) ;
D14m(m(x;y;z );t;m(x;v;z )) =m(m(z;t;x);y;m(z;v;x)) ;
Trevisan [1951], who suggested the basis
D01
6=fD6;D8;D9;D15g;
where
D15m(m(x;y;z );t;v ) =m(m(x;v;t);y;m(z;t;v ));
Sholander [1951], [1952], who proposed the systems
D01
7=fD16;D17g;
D01
8=fD9;D18;D19g;
(D01
7without proof), where
D16m(0;x;m(1;y; 1)) =x;
D17m(x;m(y;z;t);v ) =m(m(x;z;v );t;m(y;x;v ));
D18m(x;x;y ) =x;
D19m(m(x;y;z );t;v ) =m(m(t;v;x);m(t;v;y );z);
Wang [1953], who proposed the basis
D01
9=fD8;D9;D20g;
where
D20m(x;m(y;z;t);v ) =m(m(v;z;x);m(x;y;v );t);
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
66 3. Distributive Lattices
and Soboci nski [1962], who provided six independent systems, namely
D01
10=fD8;D9;D21g;
D01
11=fD22;D23g;
D01
12=fD21;D24g;
D01
13=fD21;D25g;
D01
14=fD8;D26g;
D01
15=fD8;D27g;
where
D21m(x;m(y;z;t);v ) =m(m(x;t;v );m(x;y;v );z);
D22m(0;m(y;x;x); 1) =x;
D23m(x;m(y;z;t);v ) =m(m(x;t;v );m(x;y;v );z);
D24m(0;m(x;x;y );1) =x;
D25m(0;m(x;y;x); 1) =x;
D26m(0;m(x;m(y;z;t);v );1) =m(m(x;t;v );m(x;y;v );z);
D27m(x;m(y;z;t);v ) =m(0;m(m(x;t;v );m(x;y;v );z);1):
Note that each of the systems D01
1 D01
15involves 5 variables. Four
systems making use of only 4 variables were given by authors who, curiously
enough, do not mention Problem 64. Namely, Kolibiar and Marcisov a [1974]
proposed the system
D01
16=fD6;D7;D28;D29g;
where
D28m(x;y;y ) =y;
D29m(m(x;y;z );t;z ) =m(x;z;m(t;z;y ));
while Za chik [1974] devised the systems
D01
17=fD6;D7;D8;D9;D30g;
D01
18=fD8;D9;D31g;
D01
19=fD25;D31g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
3.3 De ning Bounded Distributive Lattices in Terms of the Median Operation 67
where
D30m(m(x;y;z );t;x) =m(x;y;m(z;t;x)) ;
D31m(x;m(y;z;t);y ) =m(y;m(z;x;y );t):
Malliah [1979] announced without proofs many independent systems
which solve Problem 64: 313 systems of 3 axioms, 9 bases having 4 ax-
ioms and several systems containing 5 axioms, obtained by permutations
of variables. The author claimed that these systems includes all previously
known solutions to Problem 64, but the references are incomplete. As an
example, she proved that the system
D01
20=fD8;D9;D32g;
where
D32m(m(x;y;z );t;v ) =m(m(v;t;x);y;m(v;t;z ));
is independent and characterizes bounded distributive lattices. Finally the
author provided with full proofs the following 15 independent two-identity
systems:
D33m(x;m(0;y; 1);x) =x;
together with one of
D34m(0;m(x;m(y;z;t);v );1) =m(m(x;z;v );t;m(x;y;v ));
D35m(x;m(y;z;t);v ) =m(0;m(m(x;z;v );t;m(x;y;v ));1);
D36m(m(0;x; 1);m(y;z;t);m(0;v; 1)) =m(m(x;z;v );t;m(x;y;v ));
D37m(x;m(y;z;t);v ) =m(m(0;m(x;z;v );1);t;m(0;m(x;y;v );1));
D38m(m(0;x; 1);m(0;m(y;z;t); 1);m(0;v; 1))
=m(m(x;z;v );t;m(x;y;v ));
D39m(m(y;z;t);v )
=m(m(0;m(x;z;v );1);m(0;t; 1);m(0;m(x;y;v );1)) ;
then D 8together with one of
D40m(0;m(x;m(y;z;t);v );1) =m(m(x;z;t);t;m(x;y;v ));
D41m(m(0;x; 1);m(y;z;t);m(0;v; 1)) =m(m(x;z;v );z;m(x;y;v ));
D42m(m(0;x; 1);m(0;m(y;z;t); 1);m(0;v; 1))
=m(m(x;z;v );t;m(x;y;v ));
D43m(x;m(y;z;t);v ) =m(0;m(x;z;v );t;m(x;y;v );1);
D44m(x;m(y;z;t);v ) =m(m(0;m(x;z;v );1);t;m(0;m(x;y;v );1));
D45m(x;m(y;z;t);v )
=m(m(0;m(x;z;v );1);m(0;t; 1);m(0;m(x;y;v );1)) ;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
68 3. Distributive Lattices
nally
D46m(x;m(y;z;t);v ) =m(m(x;z;v );t;m(x;y;v ));
together with one of
D47m(0;m(x;y;x); 1) =x;
D48m(0;m(y;x;x); 1) =x;
D49m(0;m(x;x;y );1) =x:
Dra si ckov a [1966]*, working within algebras (L;m; 0) of type (3,0), pro-
posed a variant of system D01
1in which the invariance of mwith respect to
permutations is postulated and D 9is replaced by the weaker axiom
8x8y9um(0;x;u) = x&m(0;y;u) = y:
Sholander [1952] de ned bounded distributive lattices in terms of the
segments already mentioned in Chapter 1:
(1.4.1) [a;b] = fx2Lja^bxa_bg:
Pic [1969] made an attempt to obtain characterizations of bounded dis-
tributive lattices in terms of an n-ary operation instead of a ternary one.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Chapter 4
Boolean Algebras
Boolean algebras have a special position in lattice theory. The concept
of Boolean algebra and the more general concept of a lattice crystallized
more or less simultaneously, towards the end of the XIX-th century; cf.
historical notes in Birkho [1948]. Ever since then the theory of Boolean
algebras has tremendously developed, as well as its numerous applications
in logic, measure theory, probability theory, topology, computer science and
others. In particular there are more essentially distinct sets of axioms for
Boolean algebras than for any other class of lattices.
One of the factors which produce the diversity of the axiomatics of
Boolean algebras is the choice of the signature. In this chapter we basically
regard a Boolean algebra as an algebra (B; _;^;0;0;1) of type (2,2,1,0,0),
but other de nitions work with shorter types. Thus, a Boolean algebra is
uniquely determined by its bounded-lattice reduct (B; _;^;0;1) and even
by its lattice reduct (B; _;^); the reduct (B; _;0) determines the whole
Boolean-algebra structure as well. A Boolean algebra is also equivalent with
each of the algebras (B; ;1);(B;j), wherex y=x^y0andxjy=x0^y0,
with (B;m;0), wheremis the median operation, and with a Boolean ring.
These equivalences fall under the following general scheme.
Let and be two signatures. A class Aof algebras is de ni-
tionally equivalent to a class Bof algebras provided there is a family
of polynomials and a family of polynomials such that for every
(B;F )2 A and every (B;G) 2 B the following hold: (i) (B; (F))2 B,
(ii) (B; (G)) 2 A, and (iii) (F ) =Fand (G) = G. It follows
that the categories AandBare isomorphic. In particular the relation-
ships between a reduct and an extension mentioned above are obtained for:
; GF, then = (' g)g2with'g(F) =gfor allg2, and
= ( f)f2with f(G) =ffor allf2.
We will apply the above scheme to the class Aof Boolean algebras.
Each system of axioms de nes a class Bof algebras and it turns out
that properties (i) and = 1are immediate, so that the proof reduces
69
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
70 4. Boolean Algebras
to establishing that transforms a algebra into a Boolean algebra and
=1, which is done by using the system of axioms.
Note that de nitional equivalence is stronger than the equivalence be-
tween two axiom systems of di erent signatures, usually met in the litera-
ture. The latter concept requires only properties (i) and (ii). Yet property
=1is easily checked in most cases.
The present chapter is organized as follows. The rst three sections
present Boolean algebras within lattices. In x1 we list systems of axioms in
which complementation is not taken as one of the basic operations; instead,
an axiom states that every element has (at least) a complement and the
uniqueness of the complement must be proved. In the next two sections
complementation is taken as a basic operation. So x2 deals with Boolean
algebras expressed in terms of join, meet and complementation, while x3
presents systems of axioms in terms of join and complementation only,
which is possible due to the De Morgan law. The axiomatics of the im-
portant concept of Boolean ring is the subject matter of x4. Thus in xx2{4
Boolean algebra are de ned by associative operations, while in x5 the def-
initions use nonassociative binary operations: di erence or implication or
She er stroke. As shown in x6, Boolean algebras can also be de ned by
using ternary operations: majority decision m, or rejection m0, or in terms
ofn-ary She er functions. The tools used in x7 are the partial order and
the relation x y()x^y= 0. In x8 we present several characterizations
of Boolean algebras within uniquely complemented lattices; note that xx1,8
are an exception to the policy announced in the Introduction, namely to
omit characterizations of a class of lattices within a larger class. So this
chapter illustrates the fact that Boolean algebras can be presented as alge-
bras of various signatures; the metatheorem proved in x9 says that Boolean
algebras can be characterized by a single identity, whatever be the type.
On the other hand, Boolean algebras are related to orthomodular lattices
both by their origins (classical propositional calculus and quantum logic,
respectively) and by the interferences between their axiomatics. That is
why we have included a nal section dealing with orthomodular lattices.
In view of a uniform approach, the following condition is understood
throughout this chapter:
every system of axioms requires that the support set B of the algebra be of
cardinality at least 2,
even if the original paper does not mention this. Whenever the type of
the algebra presupposes two constants 0 and 1, this tacit condition has the
form 0 6= 1.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.1 Boolean Lattices 71
We also apply a uniform treatment to the constants 0 and/or 1 : unless
otherwise stated, we include them among the basic operations, even if the
original formulation was something like \there is an element 0 such that
x_0 =xfor allx". As a matter of fact, we already did so in the previous
chapters.
4.1. Boolean Lattices
Boolean lattices are de ned as complemented distributive lattices; they
are usually known as Boolean algebras, implying that complementation is
incorporated as a basic operation, along with join, meet, 0 and 1.
LetLbe a bounded lattice. We regard it as an algebra (L; _;^;0;1) of
type or signature (2,2,0,0). We say that an element x2Liscomplemented
if there is an element y2Lsuch thatx_y= 1 andx^y= 0; theny
is called a complement ofx. Note that 0 and 1 are complements of each
other. We denote by C(L) the set of complemented elements of L.
Proposition 4.1.1. In a bounded distributive lattice an element can have
at most one complement.
Proof: Supposeyandzare complements of x. Then
y=y^1 =y^(x_z) = (y ^x)_(y^z) = 0_(y^z)
= (x^z)_(y^z) = (x _y)^z= 1^z=z:
2
We will use the notation x0for the possible unique complement of an
elementxof a bounded distributive lattice.
Proposition 4.1.2. If L is a bounded distributive lattice, then C(L) is a
subalgebra of the algebra (L;_;^;0;1)and for every x;y2Lwe have
(1) (x_y)0=x0^y0;
(2) (x^y)0=x0_y0;
(3) x00=x:
Comment Properties (1), (2) are known as the De Morgan laws, while
(3) is called the law of double negation.
Proof: We use Proposition 1.1. Property (3) is just a restatement of the
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
72 4. Boolean Algebras
identitiesx0_x= 1 andx0^x= 0. Property (2) is the dual of (1), and
the latter follows from
(x_y)_(x0^y0) = (x _y_x0)^(x_y_y0) = 1^1 = 1;
(x_y)^(x0^y0) = (x ^x0^y0)_(y^x0^y0) = 0_0 = 0:
2
Note that a bounded lattice such that 0 = 1 is trivial, that is, it reduces
to a singleton. For x=x^1 =x^0 = 0.
By acomplemented lattice we mean a bounded lattice Lsuch that all of
its elements are complemented, that is, such that L=C(L). A non-trivial
complemented distributive lattice will be called a Boolean lattice. Applying
again Proposition 1.1, we see that a Boolean lattice can be endowed with
a unary operation0called complementation, which sends each element x
to its complement x0. This enriched structure is called a Boolean algebra.
In other words, a Boolean algebra is an algebra (B; _;^;0;0;1) of type
(2,2,1,0,0) such that its reduct (B; _;^;0;1) is a Boolean lattice and the
unary operation0is complementation.
Remark 4.1.1. As a matter of fact, in the de nition of complemented
lattices (not necessarily distributive) it suces to suppose that 0 and 1 are
two distinguished elements of L, and it will follow that they are bounds of
L. Forx_y= 1 implies x1, while from x^y= 0 we infer 0 x, and
these inequalities hold for all x.
We have thus presented the concept of Boolean algebra as it stands
nowadays. However this axiomatic way of thinking has emerged from a
more \computational" point of view of Boole himself and his direct follow-
ers, who were interested in computing with the truth values 0 and 1, or
equivalently, with sets (\classes", \Gebiete"); see e.g. Boole [1847], [1854],
Schr oder [1890-1905]. Here are the most important properties they have
found:
B_
1x_x=x;
B^
1x^x=x;
B_
2x_y=y_x;
B^
2x^y=y^x;
B_
3 (x_y)_z=x_(y_z);
B^
3 (x^y)^z=x^(y^z);
B_
4x_(x^y) =x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.1 Boolean Lattices 73
B^
4x^(x_y) =x;
B_
5x_(y^z) = (x _y)^(x_z);
B^
5x^(y_z) = (x ^y)_(x^z);
B_
6x_0 =x;
B^
6x^1 =x;
B_
7x_1 = 1;
B^
7x^0 = 0;
B_
8x_x0= 1;
B^
8x^x0= 0;
B9x00=x:
From the contemporary point of view, identities B 1 B9can be regarded
as a highly redundant system of axioms for Boolean algebras, while the
compact de nition of Boolean lattices as complemented distributive lattices
is due to Birkho [1948].
In this section we will present systems of axioms for Boolean lattices.
Note that any such system must contain (at least) an axiom which is not an
identity but states that for each element xthere is an element x0satisfying
properties B 8or some other property equivalent to B 8. Unless otherwise
stated, these systems refer to algebras (B; _;^;0;1) of type (2,2,0,0).
The rst system of axioms for Boolean lattices and for Boolean alge-
bras in general was given by Whitehead [1898]. It is worth mentioning that
his Treatise on Universal Algebra is indeed a book on universal algebra in
the modern sense of the word! In particular, at that time Boolean algebra
was \the only known member of the non-numerical genus of universal al-
gebra" as noted by Whitehead; cf. Birkho [1948], footnote 1 to Ch.X. So
Whitehead may be viewed as the founder of the modern concept of Boolean
algebra.
The system given by Whitehead is
B1=fB_
1;B^
1;B_
2;B^
2;B_
3;B^
3;B_
4;B^
5;B^
6;B10g;
where
B10 8x9x0x_x0= 1 &x^x0= 0:
To see that the algebra de ned by system B1is actually a Boolean lattice, it
remains to prove that it is a distributive lattice. This follows from Theorem
3.2.1, because the commutativity B 2enables one to write axiom B^
5in the
form D^
2, while B^
5;B^
1and B_
4imply
x^(x_y) = (x ^x)_(x^y) =x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
74 4. Boolean Algebras
that is, B^
4, which coincides with L^
4. Incidentally, we have also proved that
axioms B_
1;B_
3and B^
3are redundant.
Huntington [1904], [1932], [1933a], [1933b] proposed several systems of
axioms characterizing Boolean algebras; those given in [1904], [1933a] are
known as Huntington's rst-sixth sets of postulates for Boolean algebras.
Huntington's rst set is essentially
B2=I=fB_
2;B^
2;B_
5;B^
5;B_
6;B^
6;B10g:
To prove the correctness of system B2, it remains to show that every
algebra satisfying this system is a distributive lattice. In view of Sholander's
Theorem 3.2.1, it suces to prove that B2implies B^
4. But
x^(x0_0) = (x ^x0)_(x^0) = 0 _(x^0) =x^0;
x^(x0_0) =x^x0= 0;
hencex^0 = 0, therefore
x^(x_y) = (x _0)^(x_y) =x_(0^y) =x_0 =x:
The original formulation of this self-dual system refers to an algebra
(B;_;^) of type (2,2): instead of B 6one postulates the existence of 0 and
1 satisfying B 6, while the last axiom requires B 10provided the elements 0
and 1 exist and are unique. Under this form, the independence of the system
was proved by Huntington himself and reproved by Bernstein [1924] and
later by Gerrish [1978], after a careful discussion, but apparently unaware of
of Bernstein's proof. The same problem was tackled by R uthing [1974]*; cf.
MR 58(1979), #425. Sampathkumar [1967]* provided a version of system
B2; cf. MR 50(1975), #12846. Diamond [1934a] realized the complete
existential theory of the system.
It is Moore [1910] who de ned the complete existential theory of a system
ofnconditionsp1;:::;p n. By this term he meant the description of all
the implications that hold between Boolean combinations of p1;:::;p n. In
particular if there is no such implication, the system is said to be completely
independent. Several complete existential theories occurring in the theory
of partially ordered sets, set theory, lattice theory and the theory of Boolean
algebras have been studied so far; see Appendix F.
Successive works of Del Re [1911]*, Bernstein [1914]* and Stone [1935b]
yielded the independent system
B3=fB_
1;B_
2;B^
2;B^
5;B_
6;B^
6;B10g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.1 Boolean Lattices 75
which is simpler than a previous system given by Del Re [1907].
Stone [1935b] devised the system
B4=fB_
1;B^
1;B_
2;B^
5;B_
6;B^
11;B10g;
where
B^
11 (x_y)^z= (x^z)_(y^z):
Newman's sets of axioms [1941] are essentially
B5=fB^
1;B^
5;B^
11;B_
12;B10;B_
13g;
B6=fB^
1;B^
5;B^
11;B^
12;B10;B_
13g;
B7=fB^
5;B^
11;B_
12;B^
12;B10;B_
13g;
where
B_
12 0_x=x;
B^
12 1^x=x;
B_
13 (z_z)_x=x8x=)z_z=x8x:
In their original forms, systems B5 B7are independent.
The next three de nitions are free from postulated special elements 0,1,
meaning that not only the algebra is of type (2,2), but 0 and 1 do not occur
in the axioms, their existence being proved from the axioms.
It is Bernstein [1915-16] who introduced the axiom
B14 8x9x08y y_(x^x0) =y&y^(x_x0) =y
and devised a self-dual 5-axiom system for Boolean lattices. As shown by
Montague and J. Tarski [1954], one of the two axioms B 2in that system
is redundant and by deleting it one obtains an independent system, e.g.,
B8=fB_
2;B_
5;B^
5;B14g:
Diamond [1934b] provided the independent self-dual system
B9=fB_
15;B^
15;B14g;
where
B_
15 (y^z)_x= (z_x)^(y_x)
and B^
15is the dual axiom. Sholander obtained an even simpler system:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
76 4. Boolean Algebras
Theorem 4.1.1. (Sholander [1951]) The system
B10=fB^
4;B^
16;B17g;
where
B^
16x^(y_z) = (z ^x)_(y^x);
B17 8x9x08y y^(x_x0) =y_(x^x0)
characterizes Boolean algebras and is independent.
Proof: First note that Bis a distributive lattice by Theorem 3.2.1. Then
B17implies that yx^x0andx_x0yfor anyy, showing that x^x0= 0
(i.e., least element) and x_x0= 1 (i.e., greatest element).
To prove the independency consider e.g. by the following models. For
B^
4take the set f0;1gendowed with the constant operations x_y=x^y=
1. For B^
16take the non-modular lattice N5, change the notation to o=
0; x=a; y=b; z=c; u= 1 and de ne 00= 1; a0=b0=c; c0=a;10= 0;
thenx_x0= 1 andx^x0= 0 hold for all x, hence B 17holds. For B 17take
any totally ordered set with more than two elements. 2
Starting from Huntington's rst set, Van Albada [1964] derived 16 inde-
pendent systems of axioms for Boolean lattices (within algebras (B; _;^)),
described by the following common wording:
A1. There is a one-sided identity 0 with respect to ^.
A2. There is a one-sided identity 1 with respect to _.
A3.^is one-sided distributive over _.
A4._is one-sided distributive over ^.
A5. A1 & A2 imply that one of the two identities is two-sided.
A6. A3 & A4 imply that one of the operations is two-sided distributive
over the other.
A7. A1 & A2 imply that 0 and 1 can be chosen so that 8x9y x_y=
1 &x^y= 0.
It is understood that the choice is the same in A1 as in A2 and in A3 as
in A4, so there are 16 instances of the above system A1-A7. Half of these
speci c systems are actually given in the paper with proofs, while the other
half consists of their duals.
We conclude this section with the following natural problem: given a
lattice which turns out to be the reduct of a Boolean algebra, is it true that
that Boolean algebra is unique? The armative answer follows from the
following more general theorem established by Wiener [1917]: the Boolean
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.2 Boolean Algebras in Terms of the Operations _;^and077
algebras (B; [;\;; ;! ) de ned on a Boolean algebra (B; _;^;0;0;1) by
Boolean (i.e., polynomial) operations [;\;are of the form
x[y= (x^y)_(a0^(x_y));
x\y= (x^y)_(a^(x_y));
x=x0;
= 0; ! =a0;
and they are isomorphic. See also Rudeanu [1974], Ch.12.
4.2. Boolean Algebras in Terms of the Operations
_;^and0
We begin this section with a kind of pattern for the passage from x1 tox2:
an axiom system given by Bernstein [1950], which is the transform in terms
of_;^;0of the earlier system B8of the same author [1915/6] for Boolean
lattices.
Then we give several sucient conditions ensuring that a lattice (dis-
tributive lattice) endowed with a unary operation0, or ortholattice, or New-
man algebra, is a Boolean algebra. Such results generate axiom systems for
Boolean algebras from axiom systems for the larger class.
The last part of this section includes an independent self-dual system.
In the previous section we have seen that Bernstein [1915/6] devised a
system of axioms in which 0,1 and complementation were not taken as ba-
sic operations. He then [1950] transformed the original system by working
with algebras (B; _;^;0) of type (2,2,1). So he replaced axiom B 14by
B_
18x_(y^y0) =x;
B^
18x^(y_y0) =x:
Like for his earlier system, he believed that system
fB_
2;B^
2;B_
5;B^
5;B_
18;B^
18gwas independent, which is false, as shown by
Montague and J. Tarski [1954]. After deletion of a redundant axiom, one
obtains the independent system
B11=fB_
2;B_
5;B^
5;B_
18;B^
18g:
More generally, Sioson [1964] found all the independent systems de n-
ing Boolean algebras that can be made out of the eight axioms
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
78 4. Boolean Algebras
B_
2;B^
2;B_
5;B^
5;B_
18;B^
18,
B_
19x_(y_y0) =y_y0;
B^
19x^(y^y0) =y^y0:
There are 10 such systems, grouped into pairs of dual systems; none of
them is self-dual. Then Sioson [1967] extended this research by adding the
axioms B_
1;B^
1;B_
4;B^
4.
Proposition 4.2.1. A lattice endowed with a unary operation0is a Boolean
algebra if and only if it satis es
B20 (x^y)_(x^y0) = (x _y)^(x_y0):
Proof: Since in every Boolean algebra
B_
21 (x^y)_(x^y0) =x;
B^
21 (x_y)^(x_y0) =x;
it remains to prove suciency. But B 20implies B 21because in any lattice
(x^y)_(x^y0)x(x_y)^(x_y0):
Now takex;y;z such thatx_z=y_zandx^z=y^z. Then
x= (x^z)_(x^z0) = (y ^z)_((x_z)^(x_z0)^z0) = (y ^z)_((x_z)^z0)
= (y^z)_((y_z)^z0^(y_z0)) = (y ^z)_(y^z0) =y;
therefore the lattice is distributive by Corollary 3.1.3. This implies
x^(y_y0) = (x ^y)_(x^y0) =x;
showing that y_y0= 1 is greatest element and similarly x^x0= 0 is least
element. 2
Corollary 4.2.1. Boolean algebras are characterized by the independent
self-dual set of three identities L_
62, L^
62andB20.
Proof: The system characterizes Boolean algebras by the above Propo-
sition and Theorem 1.2.5.
The independence of axiom B 20is shown by any lattice with 0 in which
y0= 0 for ally. Now, in view of duality, it suces to prove the independence
of L_
62.
Consider the set f0;1;2gendowed with the operations ^;_;0given by
x^0 = 0; x ^1 =x; x^2 = 2;
0_0 = 0; x _1 = 1_x=x;0_2 = 2_0 = 1; 2_2 = 2;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.2 Boolean Algebras in Terms of the Operations _;^and079
00= 1; 10= 2; 20= 1:
Axiom L_
26fails fory:= 0;u := 1;w := 2, for which 1 ^((v^0)_(0^2)) =
1, hence the left-hand side of L_
26is 1.
Taking in turn y:= 0; 1;2, axiom B 20reduces tox_0 = 0_x;x_2 =x_2
(because 1 ^a=a) andx_2 = 2_x, respectively.
It remains to check L^
26. Taking in turn y:= 0; 1;2, we obtain ((x _y)^
y)_(z^y) =y, therefore L^
26reduces to
L0
26y^(u_t) =y;wheret= (v_y)^(y_w):
L0
26is readily checked for y:= 1.
Further note that v_0;0_w2 f0; 1gandf0;1gis a subalgebra of our
structure, hence for y:= 0 we have t2 f0; 1g, therefore u_t2 f0; 1g(easy
remark), hence 0 ^(u_t) = 0.
Similarly,v_2;2_w2 f1; 2gandf1;2gis a subalgebra, hence for y:= 2
we havet2 f1; 2g, therefore u_t2 f1; 2g, hence 2 ^(u_t) = 2. 2
Proposition 4.2.2. A lattice L endowed with a unary operation0is a
Boolean algebra if and only if it satis es
B_
22 ((x_y0)^y)_(x^y0) =x;
B^
22 ((x^y0)_y)^(x_y0) =x:
Proof: Taking in B^
22the meet of each side with y, we get (x _y0)^y=
x^y, therefore B_
22reduces to B_
21. We obtain similarly B^
21, therefore B 20
holds and we apply the previous proposition. 2
Propositions 2.3 and 2.4 below characterize Boolean algebras among
the class of all lattices of type (2,2,1) using the idea of the uniqueness of
Mal'cev terms (see D. Kelly and R. Padmanabhan [2007] for more details).
Proposition 4.2.3. A lattice endowed with a unary operation0is Boolean
if and only if it satis es the self-dual identity
B_^
1 (x^y^z)_(x^y0^z0)_(x0^y^z0)_(x0^y0^z)
= (x_y_z)^(x_y0_z0)^(x0_y_z0)^(x0_y0_z):
Proof: In a Boolean algebra the left-hand side of B_^
1is clearlyx+y+z,
while the right-hand side equals
(x_(y^z0)_(y0^z))^(x0_(y^z)_(y0^z0)) = (x _(y+z))^(x0_(y+z)0)
= (x^(y+z)0)_(x0^(y+z)) =x+y+z:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
80 4. Boolean Algebras
Conversely, suppose identity B_^
1holds. Taking z:=yand using well-
known lattice identities, we obtain
x(x^y)_(x^y0) = (x _y)^)x_y0)x;
hence identities
(x^y)_(x^y0) =x;
(x_y)^(x_y0) =x;
hold, therefore the lattice is a Boolean algebra by Proposition 2.1. 2
Proposition 4.2.4. A lattice endowed with a unary operation0is a Boolean
algebra if and only if it satis es the self-dual identity
B_^
2 (x_(x_z)0)^(y_(y_z)0)^(x_z)
= (x^(x^z)0)_(y^(y^z)0)_(x^y):
Proof: In a Boolean algebra we have
x_(x_z)0=x_(x0^z0) =x_z0
and dually, therefore identity B_^
2follows by Corollary 3.1.1.
Conversely, suppose identity B_^
1holds. Taking z:=xand using well-
known lattice identities, we obtain
y(y_(y_x)0)^(x_y) = (y ^(y^x)0)_(x^y)y;
therefore
(1) (y_(y_x)0)^(x_y) =x;
(2) (y^(y^x)0)_(x^y) =y:
Besides, taking x:= 1; z:=xin B_^
2we get
y_(y_x)0=x0_(y^(y^x)0)_y=x0_y
andy^(y^x)0=x0^y, hence (1) and (2) reduce to B^
21and B_
21(with
xandyinterchanged), which imply B 20, therefore the lattice is a Boolean
algebra by Proposition 2.1. 2
Remark 4.2.1. A distributive lattice is a Boolean algebra if and only if it
satis es
B_
23x_x0=y_y0;
B^
23x^x0=y^y0;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.2 Boolean Algebras in Terms of the Operations _;^and081
becausex_x0andx^x0are constants and in view of Remark 4.1.1 we can
safely denote them by x_x0= 1 andx^x0= 0.
An ortholattice is a complemented bounded lattice (not necessarily dis-
tributive) satisfying the De Morgan laws and the law of double negation.
An ortholattice is a Boolean algebra if and only if it satis es B_
21. This led
to the following characterization of Boolean algebras:
Theorem 4.2.1. (Beran [1982]) The following system characterizes
Boolean algebras:
B12=fB_
18;B_
24;B_
25g;
where
B_
24 (x_y)_z= (z0^y0)0_x;
B_
25 (x^(y_z))_(x^y0) =x:
Proof: Clearly the axioms B12are ful lled in a Boolean algebra. Con-
versely, note that (x ^(x_z))_(x^x0) =x^(x_z) by B_
18and
(x^(x_z))_(x^x0) =xby B_
25, thereforex^(x_z) =x. But this identity
together with B_
18and B_
24characterize ortholattices; cf. Beran [1976]. On
the other hand, taking z:=y^y0in B_
25and using again B_
18, we obtain
the identity B_
21, which characterizes Boolean algebras within ortholattices.
2
The following generalization of Boolean algebras was introduced by
Newman [1941]: an algebra (A; _;^;0;1) of type (2,2,0,0) satisfying the
axioms B^
5;B_
6;B^
6;B10and 0 _x=x; the element x0in B10is in fact
uniquely determined by x. This structure is known as a Newman algebra
and it is isomorphic to the Cartesian product of the subalgebra of the el-
ements satisfying x_x=xand the subalgebra of the elements satisfying
x_x= 0; the former subalgebra is a Boolean algebra. So Boolean algebras
coincide with Newman algebras satisfying the identity x_x=x.
There are many axiom systems for Newman algebras; see e.g. G.D.
Birkho and G. Birkho [1946], Wooyenaka [1964], Sioson [1965a], [1965b],
[1967], Soboci nski [1972c], [1972d], [1972e], [1973]. Sioson [1967] has
proved that each of the following identities characterizes Boolean algebras
among Newman algebras: B_
1;B_
4;B^
4;B_
5;B_
19. As a matter of principle,
this generates even more systems of postulates for Boolean algebras; some
of them have been actually written down. We refer to these systems here
because they characterize Boolean algebras within a larger class, but note
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
82 4. Boolean Algebras
that from the point of view of axiom B 10they should have been presented
inx1. The rst system obtained under the in
uence of Newman algebras
is due to Birkho [1948]:
B13=fB_
1;B_
5;B^
5;B_
6;B_
12B^
12;B_
8;B^
8g:
A quite di erent idea for generating a system of axioms was suggested by
G.D. Birkho and G. Birkho [1946]. They noted that the axioms obtained
from B^
5;B^
6;B_
8by applying the transformations of the 8-element group
generated by left-right symmetry and duality form a (redundant) system
de ning Boolean algebras. This is related to their philosophic view that
\the nal form of any scienti c theory T is (1) based on a few simple
postulates, and (2) contains an extensive ambiguity, associated symmetry,
and underlying group G, in such wise that … T appears nearly self-evident
in view of the Principle of Sucient Reason".
The problem of nding an independent self-dual system of identities for
Boolean algebras was raised by Gr atzer [1971], Problem 29 (Huntington's
rst system and Diamond's system B9are not equational). The rst answer
to this problem was given in
Theorem 4.2.2. (Padmanabhan [1983]) The self-dual system
B14=fB_
26;B^
26;B_
27;B^
27;B_
8;B^
8g;
where
B_
26 (x^y)_y=y;
B^
26 (x_y)^y=y;
B_
27x_(y^z) = (y _x)^(z_x);
B^
27x^(y_z) = (y ^x)_(z^x);
characterizes Boolean algebras and is independent.
Proof: Clearly the well-known proof that the absorption laws imply the
idempotency laws (see e.g. Ch.1,×1, proof for L1) works also for the form
B26of absorption. Now B^
1;B_
27and B^
1imply
x_y=x_(y^y) = (y _x)^(y_x) =y_x;
and similarly x^y=y^x. Therefore axiom B^
26coincides with B^
4, while the
distributivity B^
27can be written in the form x^(y_z) = (z ^x)_(y^x).
This identity and B^
4ensure that the algebra is a distributive lattice by
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.2 Boolean Algebras in Terms of the Operations _;^and083
Sholander's Theorem 3.2.1, therefore it is a Boolean algebra by Remark
1.1.
The independence is shown by the following models. For B_
26take the
setf0;1gwith the operations x_y= 1; x^y=y; x0= 0. For B_
8take
the chain f0;1gwith its lattice operations and de ne x0= 0. For B_
27take
a chain f ;!g, where <! , and de ne x_y=y; x^y= min(x;y ); x0=
;0 = 1 = . For the dual axioms take the dual models. 2
The fact that in lattice theory each distributive law implies the other
makes it hard to produce an independent basis containing both distributive
laws. Diamond's self-dual independent system B9realizes this by axioms
B_
15and B^
15, but uses axiom B 14, which is not an identity. If we replace
B14by the corresponding identities
B_
14x_(y^y0) =x;
B^
14x^(y_y0) =x;
we obtain the following variant of Diamond's system:
Theorem 4.2.3. The set fB_
14;B^
14;B_
15;B^
15gis an independent self-dual
basis for Boolean algebras.
Proof: Suppose an algebra (L; _;^;0) satis es the above system.
By using in turn B_
14;B^
15and again B_
14, we obtain
x^y= (x^y)_(u^u0) = (y _(u^u0))^(x_(u^u0)) =y^x
and similarly x_y=y_x, so that in the following we can freely use
commutativity.
It follows by B_
14;B_
15and B_
14that
(3)x^(y^y0) = (x ^(y^y0))_(x^x0) =x^((y^y0)^y)
=x_(y^y0) =x:
Now it follows by B_
14;B^
15,(3) and B_
14, that
x^(x_y) = (x _(y^y0))^(x_y) =x_((y^y0)^y) =x_(y^y0) =x
and similarly x_(x^y) =x. Therefore the reduct (L; _;^) is a distributive
lattice by Sholander's Theorem 3.2.1. Now axioms B_
14and B^
14ready^y0
xy_y0, showing that y^y0andy_y0are the least and greatest elements
of the lattice L, hence they are constants and by setting y^y0= 0 and
y_y0= 1 we get a Boolean algebra (L; _;^;0;0;1).
To prove the independence of the system, let (B; _;^;0;0;1) be an arbi-
trary Boolean algebra. Set x y=x^y0andxy = (x^y)_(x0^y0).
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
84 4. Boolean Algebras
Consider the algebra (B; 0; ;0) of type (2,2,1) (so x_y= 0 for all
x;y). This algebra does not ful l B_
14because 0 6= 1, while the other
axioms aresatis ed because x 0 =x;0 = 0 0 and 0 x= 0.
The algebra (B; _;;0) of type (2,2,1) does not ful l B_
15because (x _
y)0 =x0^y0buty0)_(x0) =x0_y0, while the other axioms are
satis ed because
x_(yy0) =x_0 =x;
x(y_y0) =x1 =x;
(xy )_z= (x^y)_(x0^y0)_z ;
(y_z)(x_z) = ((y _z)^(x_z))_((y0^z)^(x0^z0))
= (x^y)_z_(x0^y0^z0) = (x ^y)_z_(x0^y0):
The duals of the above models establish the independence of the axioms
B^
14and B^
15, respectively. 2
We conclude with a few systems of axioms that are expressed in terms
of_;^;0but do not fall under the general ideas emphasized above.
Ponticopoulos [1962] suggested a system consisting of B_
2;B^
2;B9and
an additional very long axiom.
Sampathkumar [1963] provided the system
B15=fB_
2;B^
2;B^
5;B^
6;B_
7;B_
8;B^
8g;
Carloman [1976] proposed the system
B16=fB_
2;B_
4;B_
5;B_
8g;
while Lisovik [1977] devised the system
B17=fB_
2;B_
5;B^
28;B_
29g;
where
B^
28x^y= (x0_y0)0;
B_
29 (y^y0)_x=x:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.3 Boolean Algebras in Terms of the Operations _and085
4.3. Boolean Algebras in Terms of the Operations _and0
The rst de nition of Boolean algebras in terms of join and complements
is due to Huntington [1904] and is known as Huntington's third set. In
modern terms, he considered a bounded join semilattice satisfying two more
postulates: 1) for each element xthere is an element x0such thatx_x0= 1
and the unique lower bound of xandx0is 0, and 2) every two non-0 elements
xandyhave a lower bound z6= 0.
The other systems of axioms presented in this section refer to algebras
Bendowed with a binary operation _and possibly with a constant 0 or a
constant 1 or both. Some of these systems include a subset equivalent to
the fact that (B; _) is a semilattice, others do not; in all cases the operation
_is taken to be the join semilattice of the Boolean algebra. If the type
of the original algebra includes only one of the constants 0,1, the other
constant is de ned by complementation. If the type of the algebra is just
(2,1), the elements 0 and 1 are obtained as in Remark 2.1 or alike. In all
cases the meet operation is de ned byy
(1) x^y= (x0_y0)0;
the point is that in a Boolean algebra identity (1) holds by (1.2) and B 9.
The rst equational de nitions of Boolean algebras are again due to
Huntington [1933a] and known as his fourth and fthsystems, namely the
independent systems
B18=IV=fB_
2;B_
3;B_
30g;
where
B_
30 (x0_y0)0_(x0_y)0=x;
and
B19=V=fB9;B_
31;B_
32g;
suggested by She er's system B52, where
B_
31x_(y_y0)0=x;
B_
32 ((y0_x)0_(z0_x)0)0=x_(y_z)0:
Of course, the commutativity B_
2is obtained due to the \reverse"
order in the left-hand side of B_
32. However we can prove commuta-
tivity even without B_
31and with the \normal order" in B_
32, provided
two other simple axioms are introduced. This is shown by the system
yNote that (1) is a de nition, not to be confused with Lisovik's axiom B^
28.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
86 4. Boolean Algebras
B0
20=fB_
6;B9;B_
33;B_
34g, where
B_
33x_x0= 00;
B_
34 ((x_y0)_(x_z0)0)0=x_(y_z)0:
Rudeanu [1963], following a suggestion of Moisil, proved that system
B20=B0
20[ fB_
2g
characterizes Boolean algebras, as well as the independence of the axioms
inB20, and asked whether B_
2can be deduced from B20. Petcu [1967]
answered in the armative.
On the other hand, the commutativity of the join can replace the law
of double negation in Huntington's fth set:
Proposition 4.3.1. (Lisovik [1997]) The system
B21=fB_
2;B_
31;B_
32g
characterizes Boolean algebras and is independent.
Proof: It suces to derive B 9fromB21. First we note that
((x00_x)0_(x000_x)0)0=x_(x0_x00)0=x;
showing that every element xcan be written in the form x=w0. Now set
(x_x0)0=z. Using freely the commutativity, we compute
x=z_x=z_w0=z_(z_w)0
= ((z0_z)0_(w0_z)0)0= ((w0_z)0)0=w000=x00:
The independence is left to the reader. 2
The fact that Boolean algebra is the algebraic counterpart of classical
propositional calculus raises the possibility of transforming systems of ax-
ioms for the latter theory into alternative de nitions of Boolean algebras.
For instance, Bernstein [1931] transcribed a well-known formal system of
propositional calculus into the language of an algebra (B; _;0;1), e.g.
p= 1 &p0_q= 1 =)q= 1
was the transcription of modus ponens, while the axiom `p!(p_q)
becamep0_(p_q) = 1; etc. Henle [1932] noted that the system obtained in
this way did not suce to characterize Boolean algebras. Then Huntington
[1933a] corrected the error and obtained an independent system known as
hissixth set. Bennett [1933] simpli ed one of the axioms and obtained a
new independent system.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.3 Boolean Algebras in Terms of the Operations _and087
Another system originating in propositional calculus was given by Lewis
and Langford [1932], namely
B22=fB_
1;B_
2;B_
3;B_
7;B_
35;B_
36g;
which de nes Boolean algebras as join semilattices with greatest element 1
and a unary operation0satisfying
B_
35x_y0= 1 =)x_y=x;
B_
36x_y=x&x_y0=x=)x= 1:
However Rudeanu [1962] observed that axiom B_
7is redundant (because
takingx:= 1_10; y:= 1 in B_
36yields 1 _10= 1, then the same axiom
withx:=x_1; y= 1 implies x_1 = 1), while the system
B23=fB_
1;B_
2;B_
3;B_
35;B_
36g
is independent.
In Jarbuch Fortschritte Math. 59(1933), p.59, it is stated that Hunt-
ington [1933b]* improved an axiom system given by Lewis and Langford.
The following important theorem is due to Frink [1941], with a proof
which invokes the axiom of choice and Zorn's lemma. An elementary proof
was given by Padmanabhan [1981], which is essentially the one given below.
Theorem 4.3.1. An algebra (B;_;0)can be made into a Boolean algebra
if and only if (B;_)is a semilattice which satis es
B_
37x_y=y()x0_y=z0_z:
Proof: Sincex0_x=z0_zby B_
37, it follows by Remarks 2.1 and 1.1
that 1 =x_x0is greatest element, hence B_
37reads
(2) x_y=y()x0_y= 1:
Takingy:=x00, this implies further
(3) x_x00=x00
and in particular x0_x000=x000, hence
x000_x=x_x000=x_x0_x000= 1_x000= 1;
therefore (2) implies x00_x=x; comparing to (3), we obtain B 9.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
88 4. Boolean Algebras
De ne 0 = 10andx^y= (x0_y0)0. This immediately implies, via B 9,
the De Morgan laws, whence it is readily checked that ^is a semilattice
operation as well, and using also (2) we get
x^y=x() (x^y)0=x0()x0_y0=x0()y0_x0=x0
()y00_x0= 1()x0_y= 1()x_y=y;
therefore (B; _;^) is a lattice with order relation
(4)xy()x^y=x()x_y=y()x0_y= 1()x^y0= 0:
Besides,x_x0= 1, hence x^x0=x0^x= 10= 0, therefore Bis a
complemented lattice by Remark 1.1.
To prove distributivity we note that
x^(x^y)0^y00=x^y^(x^y)0= 0;
whence (4) implies x^(x^y)0y0. We obtain similarly x^(x^z)0z0,
therefore
x^(x^y)0^(x^z)0y0^z0= (y_z)0
and taking the meet of each side with y_zwe obtain
x^(y_z)^((x^y)0^(x^z)0)00= 0;
hence
x^(y_z)((x^y)0^(x^z)0)0= (x^y)_(x^z)
and since the converse inequaliy holds in any lattice, it follows that x^(y_
z) = (x ^y)_(x^z). 2
This theorem has many consequences. Note rst that since universal
quanti ers are understood in B_
37, this axiom says in fact the following:
eitherx_y=yis true and x0_y=z0_zis true for all z,
orx_y=yis false and x0_y=z0_zis false for all z.
Now the following pair of axioms has the same meaning:
B_
38x_y=y=) 8z(x0_y=z0_z);
B_
39 9z(x0_y=z0_z) =)x_y=y:
Byrne [1946] proved that the idempotency of the operation _can be
dropped in Frink's theorem, while Goodstein [1963] re-proved this resulty,
working with axioms B_
38;B_
39; he provided the dual of the system
B24=fB_
2;B_
3;B_
38;B_
39g:
yWith no reference to Frink or Byrne.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.3 Boolean Algebras in Terms of the Operations _and089
Rudeanu [1962], analyzing Byrne's proofs, observed that in fact they
constructed the duals of the systems B24and
B25=fB_
38;B_
39;B_
40g;
where
B_
40 (x_y)_z= (y_z)_x:
We give below some other corollaries of Frink's theorem. Their proofs
have a common scheme. First one shows that Bis a semilattice having as
greatest element the constant z_z0= 1. Then one checks B_
37in the form
(2).
Theorem 4.3.2. An algebra (B;_;0)can be made into a Boolean algebra
if and only if (B;_)is a semilattice which satis es B9;B_
23and
B_
41 (x0_y)0_(x_y)0=y0:
Proof: We setx_x0= 1, which is a constant by B_
23. Then taking
y:=x0in B_
41, seting 10= 0 and using B 9we obtainx_0 =x. Besides,
x_1 =x_x_x0=x_x0= 1. Now (2) follows easily via B_
41and B 9:
x0_y= 1 =)0_(x_y)0=y0=)x_y=y;
x_y=y=)x0_y=x0_x_y= 1_y= 1:
2
Proposition 4.3.2. (Malliah [1968]) The following system characterizes
Boolean algebras within algebras (B;_;0;0):
B26=fB9;B_
33;B_
42;B_
43g;
where
B_
42x_(y_z) =z_(x_y);
B_
43x_y= 00=)x_y0=x:
Proof: It follows from B_
43thatx_x0= 00=)x_x00=x, whence
B_
33and B 9implyx_x=x. But this property and B_
42are S 1and S 7
in Ch.1 x1, therefore (B; _) is a semilattice and setting 00= 1, axiom B_
33
readsx_x0= 1. This implies that x1, that is, 1 is greatest element.
Now we check (2):
x_y=y=)x0_y=x0_x_y= 1_y= 1;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
90 4. Boolean Algebras
while from B_
43and B 9we get
x0_y= 1 =)y_x0= 00=)y_x00=y=)x_y=y:
2
Proposition 4.3.3. The following system characterizes Boolean algebras
within algebras (B;_;0):
B27=fB_
1;B_
44g;
where
B_
44 ((x_y)_z)_t= (y_z)_x() ((x_y)_z)_t0=u_u0:
Proof: It follows from B_
1that
((x_x)_x)_x=x= (x_x)_x;
therefore B_
44impliesx_x0=u_u0, showing that x_x0= 1 is a constant.
Now taking t:= (x_y)_zin B_
44and using B_
1, we get (x _y)_z= (y_z)_x.
But this property and B_
1are S 6and S 1in Ch.1 x1, therefore (B; _) is a
semilattice. Taking y:=z:=xin B_
44, we obtain x_t=x()x_t0= 1,
which coincides with (2) due to commutativity. 2
A two-axiom equational characterization of Boolean algebras was pro-
vided by Soboci nski [1979]:
Proposition 4.3.4. The following system characterizes Boolean algebras
within algebras (B;_;0):
B28=fB_
1;B_
45g;
where
B_
45 ((z0_t)0_(z0_t0)0)_(x_y) =x_(y_z):
Proof: Since
x=x_x= ((x0_t)0_(x0_t0)0)_(x_x);
it follows that
(5) x=w_x;
where we have set (x0_t)0_(x0_t0)0=w. Now by applying in turn (5),
B_
45withx:=y:=w; z :=x, we obtain
x=w_(w_x) = ((x0_t)0_(x0_t0)0)_(w_w) =w_(w_w) =w;
that is,
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.3 Boolean Algebras in Terms of the Operations _and091
(6) x= (x0_t)0_(x0_t0)0;
which transforms B_
45intoz_(x_y) =x_(y_z). So the axioms system
fS1;S7gof semilattices is ful lled, therefore the commutativity reduces (6)
to B_
30and Huntington's fourth system B18is ful lled. 2
Another two-identity characterization of Boolean algebras was given by
Meredith and Prior [1968], namely
B29=fB_
46;B_
47g;
where
B_
46 (x0_y)0_x=x;
B_
47 (x0_y)0_(z_y) =y_(z_x):
In the early 1930s Robbins conjectured that in Huntington's fourth set
B18one can replace axiom B_
30by the following variant:
B_
48 ((x_y)0_(x_y0)0)0=x;
which became known as Robbins' axiom. So his conjecture was that system
B30=fB_
2;B_
3;B_
48g
characterizes Boolean algebras. Winker [1992] proved that this would be
true provided B29implies the solvability of the equation (x _y)0=y0.
McCune [1997], using the theorem prover Otter, solved the problem in the
armative, showing that Winker's equation has solutions. Otter's proof
contains fairly complex terms which are hard to understand or even to
print in a readable format. Dahn [1998] obtained a quite readable \anthro-
pomorphized" version of the proof. Other attempts in this direction are
referred to in Dahn's paper.
The program Otter was also used by Phillips and Vojt echovsk y [2005]*,
who gave the system
B31=fB_
48;B_
49g;
where
B_
49 (x_y)_z=y_(z_x);
and by McCune, Vero , Fittelson, Harris, Feist and Wos [2002], who found
ten single-identity characterizations of Boolean algebras. One of them is
B_
50 (((x_y)0_z)0_(x_(z0_(z_u)0)0)0)0=z:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
92 4. Boolean Algebras
4.4. Boolean Rings and Groups
The ring presentation of Boolean algebras is largely used, due to its com-
putional facilities. After several forerunners (cf. Birkho [1948], Ch.X,
footnote 3), it is Stone [1935a], [1936] who provided a clear-cut descripion
of the relationship between Boolean algebras and Boolean rings, which gen-
eralizes the relationship between the two-element Boolean algebra and the
ring of integers modulo 2.
Given a Boolean algebra (B; _;^;0;0;1), the transformation de ned
by 0 = 0;1 = 1 and
(1) x+y= (x^y0)_(x0^y);
(2) xy=x^y;
produces a Boolean ring (B;+;;0;1), that is, a ring with unit in which
every element is idempotent, i.e., identity x2=xholds.
In fact the idempotency implies that the ring is commutative and of
characteristic 2, meaning that identity x+x= 0 holds true. For (x +
y)(x+y) =x+y, hencexy+yx= 0, thenxy= yx. Taking y:=xwe
obtainx= x, therefore the previous identity becomes xy=yx.
Conversely, a Boolean ring (B; +;;0;1) is made into a Boolean algebra
(B;_;^;0;0;1) by the transformation de ned by 0 = 0 ;1 = 1 and
(3) x_y=x+y+xy;
(4) x^y=xy;
(5) x0=x+ 1:
The above transformations and satisfy conditions (i){(iii) in the
Introduction of this chapter. Moreover, the above Boolean ring is unique, to
the e ect that for a given Boolean algebra, any Boolean ring for which there
exist polynomial transformations satisfying conditions (i){(iii) is isomorphic
to the one given by (1) and (2); cf. Rudeanu [1961] and Gr atzer [1962].
To be speci c, the Boolean ring (B; ; ; ;! ) determined on a Boolean
algebra (B; _;^;0;0;1) by Boolean (i.e., polynomial) operations ; are
of the form
xy=x+y+a;
x y=xy+a(x+y);
and they are isomorphic. See also Rudeanu [1974], Ch.12.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.4 Boolean Rings and Groups 93
Consider the case when the Boolean algebra is a eld of sets
(B;[;\;C;?;S), meaning that Bis a family of subsets of a certain set
S, closed with respect to set-theoretical union, intersection and comple-
mentation, and ?;S2B. Then the operations of the associated Boolean
ring are the symmetric di erence of sets and the intersection. If the set Sis
in nite, then the nite subsets of Sform a subring which is still idempotent,
hence commutative and of characteristic 2, but has no unit. Such a ring will
be called a generalized Boolean ring. As shown by Gr atzer [1978], there is
a Stone-like correspondence between relatively complemented distributive
lattices with zero and generalized Boolean rings, x+ybeing de ned as the
relative complement of x^yin the segment [0;x _y] and, of course, for-
mula (5) is missing. Note that certain authors refer to Boolean rings and
generalized Boolean rings as Boolean rings with unit and Boolean rings,
respectively.
In the sequel we adopt the convention that binds stronger than +.
Unless otherwise stated, Boolean rings are characterized within algebras
(B;+;;0;1) of type (2,2,0,0) and generalized Boolean rings within algebras
(B;+;;0) of type (2,2,0).
The axiomatics of Boolean rings began with several systems of axioms
for generalized Boolean rings. Thus, Stabler [1941] suggested the system
B
32=fR1;R2;R0
2;R3;R4;R5;R0
5g;
where
R1x+ (y +z) = (x +y) +z;
R2 (9z x+z=y+z) =)x=y;
R0
2 (9z z+x=z+y) =)x=y;
R3x2=x;
R4x(yz) = (xy )z ;
R5x(y+z) =xy+xz ;
R0
5 (y+z)x=yx+zx
and the equational basis
B
33=fR3;R5;R6;R7;R8g;
where
R6 (x+x) +y=y;
R7x+ (y +z) =y+ (z +x);
R8x(yz) =y(zx);
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
94 4. Boolean Algebras
while Bernstein [1944] provided 9 independent equational bases, for instance
B
34=fR3;R5;R7;R8;R9g;
where
R9x+ (x +y) =y:
Of course, from every system of axioms B for generalized Boolean rings
one obtains a basis Bfor Boolean rings by adding the axiom x1 =xor
the axiom 1 x=x.
Miller [1952] constructed the independent systems
B35=fR10;R11;R0
11;R12g;
where
R10x+ (y +y) =x;
R11x1 =x;
R0
11 1x=x;
R12 ((x(yy ))z)((t+u) +v) = ((zy )x)(v +u) + ((zy )x)t;
and
B36=fR10;R11;R13;R14g;
where
R13 (x(yy ))z= (zy )x;
R14x((y +z) +t) =x(t+z) +xy:
Byrne [1951] obtained the system
B37=fR1;R15;R16;R17g;
where
R15 (x+y) +x=y;
R16x(t+yz) =xt+y(zx);
R17x(x+ 1) =y+y1;
and another system consisting of R 16and a rather long identity.
Tamura [1970] devised the system
B38=fR11;R18;R19;R20g;
where
R18x+ 0 =x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.4 Boolean Rings and Groups 95
R19 (x+x)y= 0;
R20 (x+ (ys +zt))s = (ys +xs) +t(zs);
while Isobe [1973]* discussed the case of generalized Boolean rings (cf. MR
48(1974), #2016).
The additive goup of a Boolean ring led to the concept of a Boolean
group ; by this term is meant a commutative group which satis es the iden-
tityx+x= 0. Boolean groups can be characterized by single identities
and can be used to obtain short characterizations of Boolean rings.
Thus, Bernstein [1939] provided 20 independent systems of axioms for
Boolean groups, among which 12 systems with 2 axioms each, namely
fR6;R7g;fR15;R7g;fR21;R7g;fR9;R7g;fR10;R7g;fR22;R7g;
fR10;R1g;fR6;R23g;fR6;R24g;fR6R25g;fR26;R1g;fR26;R7g;
where
R21 (x+y) +y=x;
R22x+ (y +y) =x;
R23x+ (y +z) =z+ (y +x);
R24 (x+y) +z= (x +z) +y;
R25x+ (y +z) = (x +z) +y;
R26x+y=x=)y=z+x:
Let us prove, for instance, that system fR6;R24gde nes Boolean groups.
We obtain in turn
y+z= ((x +x) +y) +z= ((x +x) +z) +y=z+y;
(x+y) +z= (y +x) +z= (y +z) +x=x+ (y +z);
y+y= (x +x) + (y +y) = (y +y) + (x +x) =x+x;
so that we can set x+x= 0 (a constant) and R 6becomes 0 + y=y.
Sholander [1953] proved that the identity
R27x+ (y + (z + (y + (z + (y +x))))) =y
characterizes Boolean groups, while Boolean rings are de ned by the system
B39=fR27;R28g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
96 4. Boolean Algebras
where
R28x+((zz )x+(x(y +z)+y(zt))) =yx+((11)x+(t(y +y)+z(yt))):
Proposition 4.1 and Theorems 4.1 and 4.2 below are due to Mendelsohn
and Padmanabhan [1972].
Proposition 4.4.1. A groupoid (B;+)is a Boolean group if and only if it
satis es the identity
R29x+ (((x +y) +z) +y) =z :
Proof: Setting
y:= ((x +t) +u) +t; z:= ((u +v) +x) +v
and using R 29, we havex+y=uandu+z=x, hence ((x +y) +z) +y=
x+y=u, so that R 29becomes
(6) x+u= ((u +v) +x) +v;
whence by pre-adding uand using again R 29we get
(7) u+ (x +u) =x:
Now it follows from R 29and (7) that
(8) z=x+ (((x +x) +z) +x) = (x +x) +z;
hencez+z=z+ ((x +x) +z) =x+x, again by (7). Therefore we can set
x+x= 0 (a constant) and property (8) reads z= 0 +z, hence (7) implies
x= 0 + (x + 0) =x+ 0.
Thus 0 is element zero with respect to addition, so that identity x+x= 0
shows that xexists for every x, namely x=x. Finally (6) and (7) imply
in turn
x+u= ((u + 0) +x) + 0 =u+x;
(x+u) +v= (((u +v) +x) +v) +v=v+ (((u +v) +x) +v)
= (u +v) +x=x+ (u +v):
2
Mendelsohn and Padmanabhan [1975] proved also that Boolean groups
have exactly six single-identity de nitions of shortest length, one of them
being R 29.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.4 Boolean Rings and Groups 97
Theorem 4.4.1. Let B be an algebra whose signature contains a dis-
tinguished binary operation, say +. Letx;y;z;x 1;x2;:::;x nbe variables
subject to the condition x;y;z 62 fx 1;:::;x ng, andw(x1;:::;x n)a word of
signature .Then the reduct (B;+)is a Boolean group and the algebra B
satis es the identity w(x1;:::;x n) = 0if and only if it satis es the identity
(9) x+ ((((x +y) +w) +z) +y) =z :
Proof: Settingy:= (((x +t)+w)+u)+tin (9) and using (9), we obtain
x+y=u, so that (9) becomes
(10) x+ (((u +w) +z) + ((((x +t) +w) +u) +t)) =z:
Settingz:= ((((u +w) +v) +w) +x) +vin (10) and using (9), we have
(u+w) +z=x, so that the left-hand side of (10) becomes x+ (x+ ((((x +
t) +w) +u) +t)) =x+uby (9), therefore (10) reduces to
(11) x+u= ((((u +w) +v) +w) +x) +v:
Settingx:= (u +w) +zin (10) and using (9), the left-hand side of (10)
becomesx+ (x + (((x +t) +w) +u) +t) =x+uby (9), therefore (10)
reduces to
(12) ((u+w) +z) +u=z:
Settingu:=win (11) and taking into account (12) we get
(13) x+w= ((((w +w) +v) +w) +x) +v= (v +x) +v:
Settingx:=win (9) and using (13) and (12) we obtain
(14)z=w+ ((((w +y) +w) +z) +y)
=w+ (((y +w) +z) +y) =w+z:
Settingu:=win (11) and using (14) and (12) yields
x+w= ((((w +w) +v) +w) +x) +v
= (((w +v) +w) +x) +v= ((v +w) +x) +v=x
and sincex+w=xwe see that (9) reduces to R 29, therefore (B; +) is a
Boolean group by Proposition 4.1. Now taking x:=y:=z:= 0 in (9) we
obtain the identity w= 0. 2
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
98 4. Boolean Algebras
Theorem 4.4.2. An algebra (B;+;;1)of type (2,2,0) is a Boolean ring
if and only if it satis es the single identity (9)with
w=x2
1+x1+x2(x3+x4)+x2x3+x2x4+x51+x5+(x6x7)x8+(x7x8)x6:
Proof: Necessity is immediate. To prove suciency, note rst that (B; +)
is a Boolean group by Theorem 4.1. In particular the operation + is asso-
ciative, so that wmakes sense.
Identityw= 0 with all xi:= 0 except x5yieldsx51 +x5= 0, that is,
x51 =x5. Hence, setting x1:= 1;x 6:=x7:=x8:= 0 inw= 0 we obtain
x2(x3+x4) =x2x3+x2x4. Thusw= 0 reduces to
x2
1+x1+ (x6x7)x8+ (x7x8)x6= 0
and puting again x6:=x7:=x8:= 0 we obtain the idempotency x2
1=x1.
Therefore (x 6×7)x8= (x 7×8)x6, and nally this implies x6x8=x8x6by
takingx7:= 1. 2
Other equational bases use the ring operations plus an extra unary
operation. Wooyenaka [1964] characterized Boolean rings in terms of the
operations +; and0. Morgado [1970] used the extra operation (unary).
By simplifying a system due to Is eki [1968], Morgado characterized Boolean
rings by the system
B40=fR0
1;R18;R30;R31g;
where
R30 ( x+x)y= 0;
R31 ((xt+yt) +zs)s =y(st) + (x(ts) + zs);
and generalized Boolean rings by the system
B
41=fR3;R0
18;R30;R32g;
where
R0
18 0 +x=x;
R32 ((xt+y) +z)s=sy+ (x(ts) +zs):
Stabler [1941] proved that the generalized Boolean rings can also be
characterized in terms of the operations _;+ by the system
B
42=fR2;R0
2;R33;R34g;
where
R33x+ ((x_y) + (x _z)) =x_(y+z);
R34 ((x_z) + (y _z)) +z= (x +y)_z:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.5 Boolean Algebras in Terms of Nonassociative Binary Operations 99
4.5. Boolean Algebras in Terms of Nonassociative Binary
Operations
The nonassociative binary operations dealt with in this section are the
di erencex y=x^y0, also known as exception operation (\x but not
y"), the implication x!y=x0_y(\ifxtheny"), the She er stroke
xjy=x0_y0and the rejection orPeirce operation xjjy=x0^y0(\neither
xnory"); the last two operations are widely used in computer science under
the names NAND (\not and") and NOR (\not or"), respectively.
Boolean algebras in terms of di erence are presented as algebras
(B; ;1) of type (2,0), the functions and referred to in the beginning
of this chapter being
(1) x0= 1 x;
(2) x^y=x y0=x (1 y);
(3) x_y= (x0 y)0;
(4) 0 = 10=x x;
and :x y=x^y0;1 = 1. It is readily checked that (F ) =F, while
for (G) = Gthe system of axioms should imply x (1 (1 y)) =x y.
The presentations in terms of implication exhibit algebras (B; !;0) of
type (2,0), with given by
(5) x0=x!0;
(6) x_y=x0!y= (x!y)!y;
(7) x^y= (x!y0)0;
(8) 1 = 00=x!x;
and :x!y=x0_y;0 = 0. Same comment as above about condition
(iii).
It is well known that Boolean algebras satisfy the principle of duality,
where Boolean duality acts as lattice duality _ $ ^;$;0$1, and
leaves the complementation0invariant. Note that x yandy!xare
dual to each other; we can say that and!areskew dual to each other.
Moreover, this remark extends to formulae (2){(4) with respect to formulae
(6){(8); for instance, the skew dual of (2) is y_x=y0!x, which coincides
with (6); etc. A practical consequence of this remark is that every system
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
100 4. Boolean Algebras
of axioms in terms of di erence can be translated into a system of axioms
in terms of implication, and conversely. In the following we use exclusively
the language of di erence and whenever the original system was in terms
of implication we mention this fact.
Unless otherwise stated, it is understood that any term of the form a0
ocurring in an axiom is a shorthand for 1 a.
The rst system of axioms in terms of di erence was given by Bernstein
[1914]*, namely the independent system
B43=fB51;B52;B53;B54g;
where
B51x (y x) =x;
B52 (x x) z= (y y) t;
B53x y=y x=)x=y;
B54 (x y) (z t) = (((x y) t0)0 ((z0 x0) y))0:
The simpler system
B44=fB55;B56;B57g;
where
B55x (y y) =x;
B56x y0=y x0;
B57 ((x y)0 (x z0))0=x (y z);
was devised by Taylor [1920a], together with its complete existential theory.
The original form of the following four systems is in terms of implication:
B45=fB58;B59;B60;B61;B62g
provided by Huntington [1932], where
B58x 1 =x x;
B59x (x y) =y (y x);
B60 (x (x y)) ((x (x y)) z) =x (x (y (y z)));
B61 (x y)0 ((x y)0 (x0 y)0)) =y;
B62x (x y0) =x y;
the independent system
B46=fB51;B63g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.5 Boolean Algebras in Terms of Nonassociative Binary Operations 101
due to Bernstein [1934], where
B63 ((x y0) (z x0))0= (x (y z)) (t t);
and the independent systems
B47=fB55;B64g;
B48=fB65;B66g;
devised by Diego and Suarez [1966], where
B64 (x0 y0) z= (y z) (x z);
B65x00 (y y) =x;
B66 (x0 y0)00 z= (y00 z) (x00 z):
G uting [1971] simpli ed system B 32, but he was unaware of systems
B33 B37. He suggested the system
B49=fB67;B68;B69;B70g;
where
B67 (x y) z= (x z) y;
B68 1 (1 x) =x(B9would be a shorthand) ;
B69x x= 10;
B70x (x y) =x y0:
The next result provides a system in which and0are basic operations,
while 0 and 1 are constructed:
Proposition 4.5.1. (Taylor [1920a]) The following system characterizes
Boolean algebras within algebras (B; ;0)of type (2,1):
B50=fB55;B71;B72g;
where
B71x x0=x;
B72 ((z x0)0 (y0 x0))0=x (y z):
Proof: B72and B 55imply ((x x0)0 (x0 x0))0=x (x x) =x, while
from B 55and B 71we get ((x x0)0 (x0 x0))0= (x x0)00=x00, therefore
x00=x. Now B 71yieldsx0 x=x0 x00=x0.
Using in turn a00=a; b0 b=b0, B72and B 71, we obtain
y0 x0= (y0 x0)00= ((y0 x0)0 (y0 x0))0=x (y y0) =x y;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
102 4. Boolean Algebras
which, due to a00=a, implies also x0 y=y0 xandx y0=y x0. Now
(x x)0= (x x)0 (y y) = (y y)0 (x x) = (y y)0;
therefore we can set (x x)0= (y y)0= 1. Then
1 x= (x x)0 x=x0 (x x) =x0;
hence the algebra (B; ;1) satis es axioms B 55and B 56(recall that in B 56,
x0is used as a shorthand for 1 x). If we succeed to prove B 57, it will
follow that the algebra (B; ;1) satis es system B44, hence it is Boolean.
But
(z x0)0 (y0 x0) = (x z0)0 (x y) = (x y)0 (x z0);
so that axiom B 72can be written in the form B 57.
Finally note that : x y=x^y0; x0=x0; : (2); (3);(4);x0=x0,
and conditions (i),(iii) are readily checked. 2
Stone [1935b] and Sampathkumar [1967]* (cf. MR 50(1975),#12846)
de ned Boolean algebras by using the restriction of the di erence x yto
the case when yx.
Is eki [1965a], [1965b], [1965c], [1972], Arai and Is eki [1965], Sicoe [1966],
Imai and Is eki [1976], characterized Boolean algebras in terms of the dif-
ferencex yand the partial order .
Bosbach [1969] provided an axiom system in terms of the di erence x y
and the join x_y. See also Bosbach [1977].
Boolean algebras in terms of the She er stroke jare algebras (B; j) of
type (2), the transformation being
(9) x0=xjx;
(10) x_y= (xjx)j(yjy);
(11) x^y= (xjy)j(xjy);
(12) 1 =xj(xjx);
(13) 0 = (x j(xjx))j(xj(xjx));
and :xjy=x0_y0, while for the algebras (B; jj) of type (2) the
transformation is
(14) x0=xjjx;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.5 Boolean Algebras in Terms of Nonassociative Binary Operations 103
(15) x^y= (xjjx)jj(yjjy);
(16) x_y= (xjjy)jj(xjjy);
(17) 0 =xjj(xjjx);
(18) 1 = (x jj(xjjx))jj(xjj(xjjx));
and :xjjy=x0^y0. So the usual duality holds between (B; M) with
(9){(13) and (B; jj) with (14){(18).
There is a pre-history of this approach, due to Stamm [1911], who con-
structed the system
B51=fBj
73;Bjj
73;Bj
74;Bjj
74;B75;Bj
76;Bj
77;Bj
78g;
where
Bj
73xjy=yjx;
Bjj
73xjjy=yjjx;
Bj
74 (xjx)j(yjz) = (x jjy)jj(xjjz);
Bjj
74 (xjjx)jj(yjjz) = (x jy)j(xjz);
B75xjx=xjjx;
Bj
76xj(xjx) =yj(yjy);
Bj
77xj(yj(yjy)) =xjx;
Bj
78 (xjx)j(xjx) =x:
She er [1913] provided the independent system
B52=fBj
77;Bj
78;Bj
79g;
where
Bj
79 (xj(yjz))j(xj(yjz)) = ((y jy)jx)j((zjz)jx)jx);
while Dines [1914/5] proved that this system is not completely independent.
Bernstein [1916] devised the independent 2-basis
B53=fBj
80;Bj
81g;
where
Bj
80 (yjx)j((yjy)jx) =x;
Bj
81 ((yj(xjx))j((zjz)j(xjx)))j((yj(xjx))j((zjz)j(xjx)))
= (xjx)j((yjy)z)):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
104 4. Boolean Algebras
Curiously enough, if one assumes that the support Bhas at least 4
elements, then the systems B52andB53become completely independent,
as was proved by Taylor [1920b] and [1917], respectively (in the latter paper
it is also proved that B53is not completely independent).
Other two-identity systems were given by Bernstein [1933], who con-
structed the basis
B54=fBj
80;Bj
82g;
where
Bj
82 (((zjz)jx)j((yjy)jx))j(((zjz)jx)j((yjy)jx)) =xj(yjz);
Meredith [1969], who obtained the system
B55=fBj
83;Bj
84g;
where
Bj
83 (xjx)j(yjx) =x;
Bj
84xj(yj(xjz)) = ((z jy)jy)jx;
and Vero [2000], who devised the system
B56=fBj
73;Bj
85g;
where
Bj
85 (xjy)j(xj(yjz)) =x:
There are also single-axiom characterizations of Boolean algebras in
terms of She er stroke.
Thus, Hoberman and McKinsey [1937] proved that an algebra (B;M)
can be made into a Boolean algebra if and only if every polynomial function
f:B !Bcan be written in the form f(x) = (f (1)^x)_(f(10)^x0),
where0;_;^;1are given by formulae (9){(12).
Sholander [1953] characterized Boolean algebras by the single axiom
Bj
86 (xj(y0jy))00= (xj(y0jz))00=)(yjx)j(z0jx) =x;
wherea0is a shorthand for aja.
McCune, Vero , Fittelson, Harris,, Feist and Wos [2002] provided an
Otter-assisted proof of the fact that the single identity
Bj
87 (xj((yjx)jx))j(yj(zjx)) =y
de nes Boolean algebras. They also proved that there is no shorter single
identity de ning Boolean algebras in terms of the She er stroke.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.6 Boolean Algebras in Terms of Ternary or n-ary Operations 105
Conjecture Bj
87has the shortest possible length (15 symbols) among the
single identities de ning Boolean algebras under any treatment.
We conclude by emphasizing that, beside nonassociativity, there is one
more link between the three operations dealt with in this section.
Note rst that, in view of duality, any system of axioms in terms of the
di erencex y/implication x!yyields a system of axioms in terms of
x_y0=y!x/ofx0^y=y x. Therefore the functions
(19) xjy; xjjy; x y; y x; x!y; y!x
share the property that each of them is a binary simple Boolean function
(i.e., a Boolean polynomial involving no constants) which can be used alone
to de ne Boolean algebras as described in the Introduction of this chapter.
For it is clear that in the case of the functions jandjjthe signature
consists of the function alone, but this is also true for the remaining four
functions (19) provided we incorporate the constant 0 or 1 into the corre-
sponding transformation ; for instance, in the case of the function x y,
x0is the polynomial 1 x; etc.
The functions (19) are the only functions with the above property.
The reason is that, as was proved by Rudeanu [1961], the functions (19)
are the only binary simple Boolean functions fsuch that every Boolean
function (i.e., Boolean polynomial) is an f polynomial. Besides, these
f polynomials involve no constants only in the case of the functions jand
jj. The latter result was rst proved by Zylinski [1925] and Lalan [1950].
Thus all the functions (19) might be called generalized She er functions.
The paper by Rudeanu [1961] studies also the even more general case when
the She er functions are not required to be simple Boolean functions, but
just Boolean functions. See also Rudeanu [1974], Ch.12.
4.6. Boolean Algebras in Terms of Ternary or n-ary Opera-
tions
In this section we survey axiomatizations of Boolean algebras in terms of
ternary majority decision, ternary rejection, conditional disjunction and of
a generalization of the She er stroke to nvariables.
We have seen in Chapter 3, x3, that bounded distributive lattices can
be characterized in terms of the median operation
(1)m(x;y;z ) = (y ^z)_(x^z)_(x^y) = (y _z)^(x_z)^(x_y):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
106 4. Boolean Algebras
This self-dual operation, also known as ternary majority decision, yields
the lattice operations via formulae
(2) x_y=m(x; 1;y); x^y=m(x; 0;y):
In particular in a Boolean algebra we have
(3) (y0^z0)_(x0^z0)_(x0^y0) = (y0_z0)^(x0_z0)^(x0_y0);
which reads
(4) m(x0;y0;z0) =m0(x;y;z );
and the function m0is known as ternary rejection.
It follows by property x00=xthat any system of axioms in terms of
majority decision can be translated into an axiom system using ternary re-
jection, and conversely. This explains the resemblance between the existing
systems of the two kinds.
Clearly, in order to de ne Boolean algebras in terms of the majority
decisionmwe also need the operation0. Although formulae (2) involve the
constants 0 and 1, one of them can be dispensed with in the signature, for
instance one de nes 1 = 00. Moreover, Grau [1947] provided a system of
axioms for Boolean algebras involving only the operations mand0without
any constant, namely
B57=fB88;B89;B90;B91;B92g;
where
B88m(x;y;m(z;t;u)) = m(m(x;y;z );t;m(x;y;u)) ;
B89m(x;y;y ) =y;
B90m(y;y;x) = y;
B91m(x;y;y0) =x;
B92m(y0;y;x) =x:
Instead of (2), Grau chooses an element p, de nes
(5) x_y=m(x;p0;y); x^y=m(x;p;y );
and proves that the resulting algebra is a Boolean algebra B(p) having p
andp0as zero and one, respectively. Moreover, the algebras B(p) andB(q)
are isomorphic for any pandq. (As a matter of fact, this isomorphism is
not limited to the algebras constructed by Grau, but it holds for all the
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.6 Boolean Algebras in Terms of Ternary or n-ary Operations 107
Boolean algebras de ned by polynomials on a given Boolean algebra, as
was proved by Goetz [1971]; see also Rudeanu [1974], Ch.12, x4).
Kalicki [1952]* proved that axiom B 90follows from the other axioms of
B57, while the system
B58=fB88;B89;B91;B92g
is independent.
We have seen in the previous chapters that changing the \natural" order
of the variables in a certain axiom may result in the creation of a shorter
axiom system. Likewise, changing the order of the variables in B 88produces
a two-axiom system for Boolean algebras in terms of mand0:
Proposition 4.6.1. (Croisot [1951]) The following system chracterizes
Boolean algebras within algebras (B;m;0)of type (3,1):
B59=fB93;B94g;
where
B93m(x;y;m(z;t;u)) = m(z;m(y;x;u);m(x;y;t)) ;
B94m(y;x;y0) =x:
Proof: We have
(50)m(x;y;x) = m(x;m(x;y;x0);m(y;x;y0))
=m(y;x;m(x;y0;x0)) =m(y;x;y0) =x;
hence
(6)m(y;x;x) = m(y;x;m(x;y;x))
=m(x;m(x;y;x);m(y;x;y )) =m(x;x;y );
so that we can prove B 89by (50), B93, (6), B 93, (50) and (50):
m(y;x;x) = m(y;m(x;x;x);m(x;x;x)) = m(x;x;m(y;x;x))
=m(x;x;m(x;x;y )) =m(x;m(x;x;y );m(x;x;x)) = m(x;m(x;x;y );x) =x:
Now B 89, B93, (6) and B 89imply
(7)m(x;y;z ) =m(x;t;m(t;z;z ))
=m(t;m(y;x;z );m(x;y;z )) =m(m(x;y;z );t;t) =m(y;x;z )
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
108 4. Boolean Algebras
provided we take t:=m(y;x;z ). Further we use B 94, (50), (7), B 93, (7),
B94, B94and obtain
m(y0;x;y) =m(y0;m(y;x;y0);m(y;x;y )) =m(y0;m(x;y;y0);m(y;x;y ))
=m(y;x;m(y0;y;y0)) =m(y;x;m(y;y0;y0)) =m(y;x;y0) =x;
that is,
(8) m(y0;x;y) =x:
Then we obtain B 92, by B 89, B94, B93, (8) and B 94:
m(y0;y;x) =m(y0;m(x;y;y );m(y;x;y0))
=m(y;x;m(y0;y0;y)) =m(y;x;y0) =x:
Finally we apply B 92, B93, (7), B 92, B92and obtain
(9)m(x;y;z ) =m(t0;t;m(x;y;z )) =m(x;m(t;t0;z);m(t0;t;y))
=m(x;m(t0;t;z);y) =m(x;z;y ):
It follows from (7) and (9) via Lemma 3.3.1 that mis invariant to any
permutation of the variables. Therefore identity (8) reduces to B 91and B 93
yields B 88because
m(x;y;m(z;t;u)) = m(x;y;m(t;z;u)) = m(t;m(y;x;u);m(x;y;z ))
=m(m(x;y;z );t;m(x;y;u)) ;
so that system B58is ful lled. 2
Note that if we wish to obtain a de nitional equivalence as described in
the beginning of this section, then we must ensure property = 1, which
amounts to the following identity:
(10) m(m(x;m(y;z;p0);p);m(y;z;p);p0) =m(x;y;z ):
Padmanabhan and McCune [1995] obtained 43 single-identity charac-
terizations of Boolean algebras in terms of the operations mand0. One of
them was constructed using a technique due to Padmanabhan and Quack-
enbush [1973], which will be described in Theorem 9.3 of this book. The
other identities were produced with the aid of the computer program Otter,
the shortest ones having 7 variables and 26 symbols each. Here they are:
B95m(m(x;x0;y);m(m(z;u;v );w;m(z;u;t)))0;m(u;m(u;m(t;w;v );z))
=y;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.6 Boolean Algebras in Terms of Ternary or n-ary Operations 109
B96m(m(x;x0y);(m(z;m(u;v;w );t))0;m(m(t;z;v );v;m(t;z;u))) = y;
B97m(m(x;x0y);(m(z;m(u;v;w );t))0;m(m(t;z;u);v;m(t;z;w ))) =y:
The de nition of Boolean algebras in terms of ternary rejection m0does
not require any more the complementation0as a basic operation. To see
this, let us re-denote rejection, say m0=. Then in every Boolean algebra
we have
(11) (x;y;z ) = (y0^z0)_(x0^z0)_(x0^y0);
(12) x0=(x;x;x);
and using (12), formulae (2) can be written in the form
(13) x_y=0(x;1;y); x^y=0(x;0;y):
Frink [1926] established a de nitional equivalence between Boolean al-
gebras (B; _;^;0;0;1) and algebras (B;; 0;1) via the transformations
de ned by (11) and de ned by (12), (13):
Proposition 4.6.2. The following system characterizes Boolean algebras
within algebras (B;; 0;1)of type (3,0,0):
B60=fB9;B98;B99;B100;B101;B102;B103g;
whose axioms, using the shortcut (12), are
B98(x;y;z ) =(y;x;z );
B99(x;y;z ) =(y;z;x);
B100(x;x;y ) =(x;x;z );
B1010(x;x0;y) =y;
B1020(x;0(y;z;t0);t) =0(0(x;y;t);0(x;z;t);t0);
B103(x;y;z ) =0(0(x0;0(y0;z0;1);0);0(y0;z0;0);1):
Proof: Properties (11){(13) and B 98{B103of a Boolean algebra are easy
to check. Therefore = 1.
Now suppose system B60is ful lled. Since (x;y;z ) =(y;x;z ) =
(x;z;y ) by B 98and B 99, the hypotheses of Lemma 3.1 are satis ed, hence
i invariant under any permutation of the variables, therefore so is 0.
Consequently, the join and meet de ned by (13) satisfy the commutativity
B2. Besides, the identity 0(0;1;y) =y, which is an instance of B 101,
implies0(1;0;y) =y, showing that identities B 6hold: 0 _y=yand
1^y=y. Likewise, B 101implies0(x;x0;0 = 0 and0(x;x0;1) = 1, which
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
110 4. Boolean Algebras
can be read in the form B 8:x^x0= 0 andx_x0= 1. Furthermore, by
applying B 102witht:= 0 we obtain
x^(y_z) =0(x;0(y;z; 1);0) =0(0(x;y; 0);0(x;z; 0);1) = (x^y )_(x^z );
that is B^
5, while fort:= 1 we get B_
5. We have thus veri ed all the axioms
of system B2.
Therefore the transformation is well de ned. Property = 1fol-
lows by B 103, which is a translation of (10). 2
The Grau-like approach applies also to ternary rejection, the role of
formulae (5) being played by
(14) x_y=0(x;p0;y); x^y=0(x;p;y );
where0is the shorthand (12).
Proposition 4.6.3. (Whiteman [1937]) The following independent system
characterizes Boolean algebras within algebras (B;) of type (3):
B61=fB99;B104;B105g;
where, using the shorthand (12), we have set
B104(x0;y;y0) =x;
B105(x;y;0(z;t;u)) =(0(x;y;z );0(x;y;t);u):
Proof: We are going to check all the axioms of Frink's system B60but
B103. This will establish the desired equivalence (but not a de nitional
equivalence).
We already have B 99. Using B 104and B 99we obtain B 9:
x00=(x000;x0;x00) =(x0;x00;x000) =x:
From B 9and B 104we deduce
(15) (x;y;y0) =(x00;y;y0) =x0:
Now B 9, (15) and B 99imply
x=x00=0(x;y;y0) =(y;y0;x);
that is B 101. Further note that B 99, (15) and B 9imply
(16) (x0;x0;x) =(x0;x;x0) =x00=x;
while from (15) in the form 0(x;y;y0) =x, B105, (16) and (15) we infer
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.6 Boolean Algebras in Terms of Ternary or n-ary Operations 111
(17)(x;x;y ) =(0(x;y;y0);0(x;y;y0);y)
=(x;y;0(y0;y0;y)) =(x;y;y0) =x0;
which implies B 100. Now a successive application of B 99, B105and B 99
yields
(x;0(y;z;t0);t) =(t;x;0(y;z;t0)) =(0(t;x;y );0(t;x;z );t0)
=(0((x;y;t);0(x;z;t);t0);
which implies B 102.
Finally we are going to prove B 98in several steps. First we observe that
B99, (17) and B 9imply
0(z;t;z ) =0(z;z;t) =z00=z;
using this, together with B 105and B 99, we obtain
(x;y;z ) =(x;y;0(z;t;z )) =(0(x;y;z );0(x;y;t);z )
=(z;0(x;y;z );0(x;y;t)) =(0(z;0(x;y;z );x);0(z;0(x;y;z );y);t);
showing that (x;y;z ) is of the form
(18) (x;y;z ) =(0(z;0(x;y;z );x);w;t):
Then, using B 99, (17) and B 9in the form
(19) 0(a;b;a) =0(b;a;a) =0(a;a;b) =a;
we compute as above
(z;0(x;y;z );x) =(x;z;0(x;y;z )) =(0(x;z;x);0(x;z;y );z)
=(x;0(x;z;y );z) =(z;x;0(x;z;y ))
=(0(z;x;x);0(z;x;z );y) =(x;z;y ):
We introduce this result into (18) with t:=0(x;z;y ) and use again (19):
(x;y;z ) =(0(x;z;y );w;t) =(t;w;t) = t0=(x;z;y ) =(y;x;z ):
We omit the proof of independence. 2
Another ternary Boolean operation is the conditional disjunction intro-
duced by Church, namely
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
112 4. Boolean Algebras
(20) x(y;z ) = (x ^y)_(x0^z);
to be read \if xthenyelsez", which is important in switching theory and
computer programming. The fact that Boolean algebras can be de ned in
terms of conditional disjunction follows from the identities
(21) x_y=x(1;y ); x^y=x(y;0); x0=x(0;1):
It was Dicker [1963] who devised the independent system
B62=fB106;B107;B108;B109;B110g;
where
B106x(y(u;v);z(u;v)) = (x(y;z ))(u;v );
B107 90 0(y;x) = x;
B108 90x(y;0) =y(x;0);
B109 90x(0;x) = 0 ;
B110 1(y;x) =y:
We only note here that the element 0 in axioms B 106{B110is unique.
For suppose 0 1(y;x) =x,x(y;02) =y(x;02) andx(03;x) = 0 3. Then
01(03;01) = 0 1and 0 1(03;01) = 0 3, hence 0 1= 03, say = 0. It follows that
02= 0(0 2;02) = 0 2(0;02) = 0.
Now let us consider the operation
(22) (x;y)z=x^y!z=x0_y0_z;
which satis es (x;x)y =x!y, hence
(23)x_y= ((x;x)y; (x;x)y )y; x0= (x;x)0;
x^y= (((x;x)y; (x;x)y )y;((x;x)y; (x;x)y ))0:
Nieminen [1976] constructed the system
B63=fB111;B112;B113;B114g;
where
B111 ((x;x)y; (x;x)y )x=x;
B112 ((x;x)y; (x;x)y ))y= ((y;y )x;(y;y)x)x;
B113 (x;x)((y;y )z) = (y;y )((x;x)z );
B114 (0;0)x= (x; 0)0:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.7 Boolean Algebras in Terms of Relations 113
In order to parry the non-associativity of the She er stroke, Moisil [1959]
introduced the She er functions of n variables,
(24) ?(x1;:::;x n) =x0
1^:::^x0
n;(n= 1;2;::: )
and their duals, which he used in the study of circuits involving electronic
tubes, transistors or cryotrons. Answering Moisil's call to an algebraic
study of She er functions of nvariables, Hammer [1959] de ned Boolean
algebras in terms of these functions. He devised the system
B64=fB?
115;B?
116;B?
117;B?
118;B?
119;B?
120;B?
121g;
where
B?
115 ?(x;x) =?x;
B?
116 ?(?? (x1;:::;x n 1);xn) =? (x1;:::;x n);
B?
117 ?(x;y) =? (y;x);
B?
118 ?(x;?(x;y)) =? (x;?y);
B?
119 ??x=x;
B?
120 ?(x;?x) =? (y;?y);
B?
121 ?(?(x;y);?(x;z)) =?? (x;?(?(?y;?z)) ;
the Boolean algebra is obtained via
(25) x_y=?? (x;y); x^y=?(?x;?y); x0=?x:
4.7. Boolean Algebras in Terms of Relations
The rst attempts to de ne Boolean algebras using the relation of partial
ordergo back to Peirce [1884] and Schr oder [1890-1905], vol.I, but these
authors didn't provide rigorous systems of postulates. Hahn [1909] devised
another unsatisfactory Schr oder-like system, and so did Pereira [1951]*,
who partially improved Schr oder's system. Church [1952] criticized the
still unsatisfactory Schr oder-Pereira system and suggested a system of his
own, in terms of both the relation and the operations _;^;0together
with 0 and 1.
As a matter of fact, it was Huntington [1904] who obtained the rst
de nition of Boolean algebras based on partial order , complementation
0and the elements 0,1, in his second system,
B65=II=fB122;B123;B124;B0
125;B1
125;B_
126;B^
126;B_
127;B^
127;B128g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
114 4. Boolean Algebras
where B 122, B123, B124are the axioms P 1, P2, P3of partial order, B 125are
the properties of 0 and 1, B 126state the existence of join and meet, and
B_
127xy&x0y=)y= 1;
B^
127yx&yx0=)y= 0;
B128x6y0=) 9z6= 0 (z x&zy):
Couturat [1914]* provided the following variant of system II:
B66=fB123;B124;B0
125;B1
125;B_
126;B^
126;B129;B10;B130g;
in which he introduced the notation x_yandx^yfor the elements described
in axioms B 126and
B129 (x_y)^z(x^z)_(y^z);
B130 160:
The elements 0 and 1 can be dispensed with, as shown by MacNeille
[1937] in his system
B67=fB123;B124;B^
126;B^
131;B^
132;B133g;
where
B^
131x^x0y;
B^
132x^yz8z=)xy0;
B133xy()y0x0:
Another way of avoiding the elements 0 and 1 is to replace them by the
unary relations Nx() 8z xzandUx() 8z zx. Byrne [1948]
de ned Boolean algebras in terms of relations NandUtogether with the
binary relation
(1) Cxy() (zx&zy=)Nz) & (x z&yz=)Uz);
which expresses the property x^y= 0 &x_y= 1, and the ternary relation
Pxyz which translates the fact that x^y=z. He devised the system
B68=fB123;B124;B134;B135;B136g;
where
B134 8x8y9p9p0Pxyp &Cpp0;
B135 9z(Cyz &Pxzp &Np) = )xy;
B136 9x9yx6y:
However a simpler system had been given ten years earlier, in terms of
the relations ;N and
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.7 Boolean Algebras in Terms of Relations 115
(2) xDy() (zx&zy=)Nz);
which is a translation of the disjointness relation xDy()x^y= 0.
Proposition 4.7.1. (Tarski [1938]) Boolean algebras are characterized by
the system
B69=fB123;B124;B_
126;B137g;
where
B137 8x9y xDy & (xDz =)zy) & (zDy =)zx):
Proof: We are checking Huntington's system II.
Takingz:=xin B137it follows that xxand so is a partial order.
For a given element x, letyandy1be two elements satisfying B 137.
ThenxDy1andxDy1=)y1y, hencey1yand similarly yy1,
showing that y=y1. So the element yin B137is uniquely determined by
xand we can set y=x0, which introduces a unary operation0. Therefore
B137can be paraphrased in the form
(3.1) xDx0;
(3.2) xDz =)zx0;
(3.3) zDx0=)zx:
De nition (2) shows that relation Dis symmetric, hence from (3.1) we
inferx0Dx, whence (3.2) implies xx00. Moreover, from x00Dx0we deduce
x00xby (3.3). Therefore x00=x.
Now we need to prove that xy=)y0x0. For suppose xy.
Supposezsatis eszxandzy0. Thenzyand sinceyDy0it follows
by (2) that Nz; this proves that xDy0, again by (2), therefore y0x0by
(3.2).
From now on we are going to use freely x00=xandxy=)y0x0.
First we denote by x_ythe least upper bound of xandy, whose
existence is stated in B_
126. Then we set x^y= (x0_y0)0. This implies
x0^y0= (x_y)0, hencex_y= (x0^y0)0. Besides,x0x0_y0= (x^y)0,
hencex^yxand similarly x^yy. Moreover, if zxandzy, then
fromx0z0andy0z0we inferx0_y0z0, hencez(x0_y0)0=x^y.
We have thus proved B^
126, where the greatest lower bound of xandyis
x^y.
In particular x^x0xandx^x0x0. ButxDx0, thereforeN(x^x0)
by (2). Sox^x0is the least element, say = 0. From x^x0= 0 we deduce
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
116 4. Boolean Algebras
x_x0= (x0^x00)0= 00. Setting 00= 1, we have x_x0= 1. Since 0 x0
for allx, it follows that x00, that is, 1 is the greatest element.
Summarizing, we have proved properties B 125. Besides, de nition (2)
readsxDy()x^y= 0. To prove B 128, we write it in the form x6y0=)
x^y6= 0. This implication follows from the fact that if x^y= 0 then
xDy, henceyDx, therefore xy0by (3.2) 2
Going back in time, we nd a paper by Yule [1926], in which Boolean
algebras are de ned solely in terms of the disjointness relation D. This is
possible because in every Boolean algebra
xy=) 8z(y^z= 0 =)x^z= 0)
=) 8z(zy0=)zx0) =)y0x0=)xy:
The system provided by Yule is
B70=fB138;B139;B140;B141g;
where
B138 8z(yDz =)xDz) &8z(xDz =)yDz) =)x=y;
B139xDx =) 8z xDz;
B140 8x9x0xDx0& (xDy =) 8z(x0Dz=)yDz));
B141 8x8y9p 8z(pDz =)qDz)()
() 8z(xDz =)qDz) &8z(yDz =)qDz):
The order relation is de ned by
(4) xy() 8z(yDz =)xDz);
which makes the meaning of the axioms quite clear.
4.8. Huntington Varieties
It has been known for a long time that in a bounded distributive lattice,
an element can have at most one complement; cf. Proposition 4.1.1. In
particular, every element of a Boolean algebra is uniquely complemented.
In 1904 Huntington conjectured the converse, that any uniquely comple-
mented lattice was distributive. In fact, the conjecture had been veri ed
for several special classes of lattice. However, in 1945 Dilworth disproved
this conjecture by proving that any lattice can be embedded into a uniquely
complemented lattice. In 1969, Chen and Gr atzer showed that this par-
ticular result can be obtained without making use of some of the more
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.8 Huntington Varieties 117
dicult machinery developed in the above paper (e.g., an extra unary op-
erator). In 1981, Adams and Sichler strenghtened the original embedding
theorem of Dilworth by showing the existence of a continuum of varieties
in which each lattice can be embedded in a uniquely complemented lattice
of the same variety. In spite of these deep theorems, it is still hard to nd
\nice" and \natural" examples of uniquely complemented lattices that are
not Boolean. The reason is that uniquely complemented lattices having a
little extra structure most often turn out to be distributive. This seems
to be the essence of Huntington's conjecture. Accordingly, we plan to at-
tack the problem backwards: that is, by nding additional (albeit, mild)
condition that, if added, would solve the problem in the armative. Many
such conditions were discovered during 1930s and 1940s. The most notable
among such conditions, due to Birkho and von Neumann, is modularity.
Let us call a lattice property P, a Huntington property if every uniquely
complemented P-lattice is distributive. Similarly, a lattice variety K is said
to be a Huntington variety if every uniquely complemented member of K is
distributive. The monograph by Salii [1988] compiles a number of known
Huntington properties. Theorem 8.1 below gives some of the important
Huntington properties that we will employ in this section to get non-trivial
axiomatizations of Boolean algebras as uniquely complemented lattices.
Remark 4.8.1. In every uniquely complemented lattice identity x00=x
holds, because both xandx00are complements of x0. Therefore condition
1 in Theorem 8.1 below is equivalent to xy()y0x0, while the two
De Morgan laws are equivalent. The following Boolean absorption laws are
also equivalent under the De Morgan laws:
BA_(x^y0)_y=x_y;
BA^(x_y0)^y=x^y:
Theorem 4.8.1. If L is a uniquely complemented lattice, then the following
statements are equivalent:
1.L is modular;
2.L has order reversible complementation, i.e., xy=)y0x0;
3.L satis es one of the De Morgan laws;
4.L satis es one of the Boolean absorption laws;
5.L satis es the Frink implication x0_y= 1 =)xy;
6.L is Boolean.
Proof: We will use freely Remark 8.1.
1 =)2: Assumeab. Modularity implies b=b^(a0_a) = (b ^a0)_a,
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
118 4. Boolean Algebras
hence
a_((a0^b)_b0) =a_(a0^b)_((a0^b)_a)0= 1 ;
by applying once again modularity we obtain
a^((a0^b)_b0) =a^b^(b0_(a0^b)) =a^((b^b0)_(a0^b)) = 0:
Since the complement of ais unique, it follows from the above two identities
thata0= (a0^b)_b0b0.
2 =)3y: Fromx^yx;ywe inferx0;y0(x^y)0, hencex0_y0
(x^y)0. Similarly, from x0;y0x0_y0we infer (x0_y0)0x^y, hence
(x^y)0x0_y0, therefore (x ^y)0=x0_y0.
3 =)5: Suppose a0_b= 1 and set X= (a0^b)_(b^(a_b0)). Then
b0_X=b0_(a0^b)_(b0_(a0^b))0= 1;
whileXb, henceb0^Xb0^b= 0. Sob0_X= 1 andb0^X=
0, whenceX=bby the uniqueness of the complement. It follows that
a0_X=a0_b= 1, that is, a0_(b^(a_b0)) = 1. Since we have also
a0^(b^(a_b0)) =a0^b^(a0^b)0= 0;
the uniqueness of the complement implies b^(a_b0) =a. Taking the meet
of each side with awe obtaina=a^b^(a_b0) =a^bb.
5 =)6: Since the converse of the Frink implication holds trivially, the
hypothesis 5 reads x0_y= 1()xy, HenceLis a Boolean algebra by
Theorem 3.1.
6 =)1 and 6 = )4: Trivial.
4 =)2: Ifxythen by taking the join of each side with x0we obtain
x0_y= 1, whence BA^yieldsx0^y0= (x0_y)^y0=y0, hencey0x0.
2
The following comments are in order. The equivalence 1 () 6 is due
to Birkho and von Neumann; see Salii [1988], page 40. Characterization
3 generalizes Theorem X.17 in Birkho [1948]. Properties 2-5 in Theorem
8.1 are in fact among the most useful computational properties of Boolean
algebras, just like B 1-B9. From another point of view, properties 2 and
5 are (Boolean) equational implications, properties 3 and 4 are Boolean
identities, while modularity is the only lattice identity in the list. There
are also several Huntington properties of a di erent nature, for instance
niteness, being atomic, having a nite order-dimension, or semimodularity.
yThis implication does not require uniqueness of the complement and is also valid in
orthocomplemented lattices; cf. Salii [1988].
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.8 Huntington Varieties 119
But so far modularity was the only known lattice identity (in terms of _
and^only) having the Huntington property.
In this section we give several new lattice identities and implications
that are Huntington properties and nally we generalize the classical result
5 of Birkho and von Neumann by providing many lattice identities which
force a uniquely complemented lattice to be Boolean.
Theorem 4.8.2. The variety generated by N5is a Huntington variety.
Proof: A specialization of the four-variable identity ( 2) in McKenzie
[1972], p.7, forming part of an equational basis for the variety N5, yields
the following identity:
(1) x^(y_z) = (x ^(y_(x^z)))_(x^(z_(x^y))):
Now leta^b0= 0. Then (1) yields
b0^(b_a) = (b0^(b_(b0^a)))_(b0^(a_(b0^b))) = (b0^b)_(b0^a) = 0
and, of course, b0_(b_a) = 1. Sob_aandbare complements of b0, therefore
b_a=b. We have thus proved that a^b0= 0 =)ab. Conversely, ab
impliesa^b0b^b0= 0, that is, a^b0= 0. So property 2 in Theorem
8.1 is ful lled, therefore the lattice is a Boolean algebra. 2
The lattice N5 is the simplest example of what is known as a splitting
lattice. It is a nite sulattice of a free lattice and it is subdirectly irre-
ducible. More generally, we can show that any variety of lattices generated
by a splitting lattice is Huntington: i.e., if a uniquely complemented lattice
belongs to a variety generated by a splitting lattice then it must be a
Boolean algebra. What really happens here is that such varieties do satisfy
a lattice equation (like (1) above) which formally implies SD_or SD^below
which, in turn, force distributivity under unique complementation. Thus
there are in nitely many lattice varieties which are Huntington in this sense;
for more details see Salii [1988] and Padmanabhan, McCune and Vero
[2007].
Theorem 4.8.3. A uniquely complemented lattice satisfying any of the two
implications below is a Boolean algebra:
SD^x^y=x^z=)x^y=x^(y_z) ;
CM_x_y=x_z=)x^((x^y)_z) = (x ^y)_(x^z):
Proof: We are going to prove order reversibility of0, which implies
distributivity by Theorem 8.1.1. Let ab. Sincea_(a0_b0) = 1, if we
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
120 4. Boolean Algebras
succeed to prove that a^(a0_b0) = 0, the uniqueness of the complement
ofawill implya0_b0=a0, that isb0a0, as desired.
Butabimpliesa^b0b^b0= 0, hence a^a0= 0 =a^b0. If SD^
holds, this implies 0 = a^a0=a^(a0_b0). Now suppose CM_holds. From
1 =a0_aa0_bwe getb_a0= 1 =b_b0, hence
b^((b^a0)_b0) = (b ^a0)_(b^b0) =b^a0
and taking the meet with aon each side we obtain
(2) a^((b^a0)_b0) = 0:
On the other hand b_a0= 1 =b_(a_b0), hence CM_implies
(3) b^((b^a0)_(a_b0)) = (b ^a0)_(b^(a_b0))
and interchanging the roles of a0anda_b0we obtain
(4) b^((b^(a_b0))_a0) = (b ^(a_b0))_(b^a0):
Using lattice absorption L^
4,ab, (4) and (3), we infer
(5)b=b^(a_a0) =b^((a^(a_b0))_a0)b^((b^(a_b0))_a0)
= (b^(a_b0))_(b^a0) =b^((b^a0)_(a_b0))b:
So all the elements occurring in (5) are equal to b. In particular b=
b^((b^a0)_a_b0), that is,b(b^a0)_a_b0. Therefore 1 = b_b0
(b^a0)_a_b0, that is,a_((b^a0)_b0) = 1. From this identity and (2)
we infer that (b ^a0)_b0is a complement of a, therefore (b ^a0)_b0=a0,
henceb0a0. 2
Theorem 4.8.4. A uniquely complemented lattice satisfying any of the the
identities
(6) x^((x^y)_((x^z)_(y^(x_z)))) = (x ^y)_(x^z);
(7) x^((x^y)_(z^(x_(y^(x_z))))) = (x ^y)_(x^z);
(8) x^((y^(x_z))_(z^(x_y))) = (x ^y)_(x^z);
is a Boolean algebra.
Proof: We check condition CM_in Theorem 8.3. Suppose x_y=x_z.
If (6) holds then
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.8 Huntington Varieties 121
(x^z)_(x^y) =x^((x^z)_((x^y)_(z^(x_z))))
=x^((x^z)_((x^y)_z)) =x^(z_(x^y)):
If (7) holds then
(x^y)_(x^z)=x^((x^y)_(z^(x_(y^(x_y)))))
=x^((x^y)_(z^(x_y))) =x^((x^y)_(z^(x_z))) =x^((x^y)_z):
For (8) see Appendix A. 2
Finally we have the promised generalization of the theorem on modu-
larity:
Corollary 4.8.1. A uniquely complemented lattice belonging to the variety
M_N5is a Boolean algebra.
Proof: Every modular lattice satis es the Huntington identity (7) be-
cause
x^((x^y)_(z^(x_(y^(x_z))))) =x^((x^y)_(z^((x_y)^(x_z))))
=x^((x^y)_(z^(x_y))) =x^(((x^y)_z)^(x_y))
=x^((x^y)_z)^(x_y) = ((x ^y)_z)^x= (x^y)_(z^x):
The lattice N5 satis es (7) as well. This can be checked by considering
the following 12 cases: x;y;z 2 f0; 1g;x=y; x =z; y =z;x;y;z in the
role of the element having 2 complements.
Therefore by Theorem 8.4 the variety M_N5 is Huntington. 2
Finally we use the Huntington property of order reversibility in order
to give one more example of a Huntington identity.
Theorem 4.8.5. The non-modular identity
(9) x^(y_(x^((y^x)_(z^(y_x))))) =x^(y_(x^z))
de nes a Huntington variety.
Proof: In view of Theorem 8.1 it suces to prove that complementation
is order-reversing. So assume ab. Note rst that a0_b= 1. Hence
takingx:=a0;y:=bin (9) yields
a0^(b_(a0^((b^a0)_z))) =a0^(b_(a0^z));
which forz:= (b^a0)0implies further
a0=a0^(b_(a0^(b^a0)0))b_(a0^(b^a0)0);
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
122 4. Boolean Algebras
whence by taking join of both sides with bwe obtainb_(a0^(b^a0)0) = 1.
Since on the other hand b^(a0^(b^a0)0) = 0, it follows that a0^(b^a0)0=b0
by the uniqueness of the complement. So b0a0, as desired.
The above identity is non-modular because it is valid in N5. 2
4.9. Boolean Algebras Are Always One-Based
In the previous sections we saw various treatments of Boolean algebras
and in most cases we could de ne the equational theory by a single axiom.
This is no accident: the equational theory of Boolean algebras has a single
axiom, whatever be the type. This follos from a more general theorem in
universal algebra, as we are going to show in this section. Firstly we recall
several prerequisites, following Gr atzer [1978], Theorems I.3.9, II.3.11 and
III.3.9.
Acongruence relation of an algebra (A;F ) is an equivalence relation on
Awhich is compatible with all the operations in F. The set Con(A) of all
the congruences on Ais made into a lattice (Con(A); \;_), where \is the
set-theoretical intersection, that is, x(\')y()xy &x'y , and_'
is the intersection of all congruences which include and'. The lattice
Con(A) is called the congruence lattice ofA. Now consider the case when
the algebra is a lattice.
Lemma 4.9.1. Ifis a congruence of a lattice, then
(1) xyz&xz =)xyz ;
(2) (x^y)(x_y)()xy =)x(x^y) &x(x_y):
Proof: (1): We have x=x^yz^y=y, henceyxz .
(2): Ifxy, thenx=x^x(x^y) and similarly x(x_y). Therefore
xy =)x(x^y) &x(x_y) =)(x^y)(x_y)
and it remains to prove the converse of the last implication. But (x ^
y)(x_y) implies (x _(x^y))(x_(x_y)), that is, x(x_y). By
interchanging xandywe obtainy(y_x), therefore xy. 2
Proposition 4.9.1. For any lattice L, the operation _of the congruence
lattice Con(L) can be described as follows: x(_')yif and only if there is
a sequence z0;z1;:::;z nsuch that
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.9 Boolean Algebras Are Always One-Based 123
(3)z0=x^yz1 zizi+1 zn=x_y
andzi([')zi+1(i= 0;1;:::;n 1);
wherea([')bmeansabora'b.
Proof: Routine. See e.g. Gr atzer, Theorem I.3.9. 2
Theorem 4.9.1. (Funayama-Nakayama) The congruence lattice of every
lattice is distributive.
Proof: Let;;' 2Con(L). It suces to prove that \(_')(\)_
(\'), since the converse inclusion holds in any lattice. Suppose x(\(_
'))y. Thenxy and letz0;z1;:::;z nbe a sequence satisfying (3). But
(x^y)(x_y) by Lemma 9.1(2), that is, z0zn, thereforezizi+1(i=
0;:::;n 1) by Lemma 9.1(1). But using also (3), we infer that for each i
we havezi(\)zi+1orzi(\')zi+1, that is,zi((\)[(\'))zi+1.
Thereforex((\)[(\'))y. 2
Recall that by an ideal of a lattice Lis meant a non-mpty subset Iof
Lsuch thatx;y2I=)x_y2Iandxy2I=)x2I. IfLis a
lattice with least element 0, it is easily seen that for every congruence of
L, the setid() =fx2Ljx0gis an ideal of L. It is also easy to see that
for every ideal Iof a distributive lattice L, the relation cn(I) de ned by
xcn(I )y() 9i2I x_i=y_i, is a congruence of L. These properties
are much strengthened in the case of a Boolean algebra.
As prescribed by universal algebra, the congruences of a Boolean alge-
bra are those congruences of the lattice reduct (B; _;^) which are also
compatible with complementation, that is, xy =)x0y0. However we
will see that Boolean congruences coincide with the congruences of the lat-
tice reduct. The ideals of the Boolean algebra are de ned as being the
ideals of the lattice reduct.
In the sequel we need the Boolean identity
(4) x_(x+y) =x_y=y_(x+y);
which follows by (4.1), B_
4and Boolean absorption.
Lemma 4.9.2. Let I be an ideal of a Boolean algebra B. Then
xcn(I )y()x+y2I :
Proof: Ifxcn(I )ythenx0^i0=y0^i0for somei2I, hencex^y0^i0= 0,
hencex^y0i2I, implyingx^y02I, and similarly x0^y2I, therefore
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
124 4. Boolean Algebras
x+y2Iby (4.1). Conversely, if x+y2Ithen it follows from (4) that
xcn(I )y. 2
Corollary 4.9.1. cn(I)is a congruence of the Boolean ring B.
Proposition 4.9.2. In every Boolean algebra the mappings 7!id()and
I7!cn(I)establish a bijection between congruences and ideals.
Proof: We already know that id() is an ideal and cn(I) is a lattice
congruence. The properties cn(id( )) =andid((I)) =Iare easily
checked using (4), Lemma 9.2 and Corollary 9.1. For instance, let us check
the former property. If xy then (x ^y0)(y^y0) = 0 and (x0^y)0,
hence (x +y)(0_0) = 0, that is, x+y2id(), therefore (4) implies
xcn(id( ))y. Conversely, the latter relation implies x_i=y_ifor some
i2id(), that is,i0. So, denoting by [a] the equivalence class of an
elementamodulo, we have [i] = [0] and
[x] = [x _0] = [x] _[0] = [x] _[i] = [x _i] = [y _i] == [y];
that is,xy. 2
Proposition 4.9.3. The congruences of every Boolean algebra (B;_;^;0;
0;1)coincide with the congruences of the lattice reduct (B;_;^)of B.
Proof: In view of Proposition 9.2, it remains to prove that the congru-
encescn(I) are Boolean. This follows from Lemma 9.2: if xcn(I )ythen
x0+y0=x+ 1 +y+ 1 =x+y2I, thereforex0cn(I)y0. 2
Corollary 4.9.2. The congruence lattice of every Boolean algebra is dis-
tributive.
Proof: By Theorem 9.1 and Proposition 9.3 2
Recall that the composition of binary relations is de ned by
x(')y() 9zxz &z'y;
and one says that and'permute, or that they are permutable, provided
'='.
Lemma 4.9.3. Let: B be a Boolean algebra, and'congruences of B,
andx;y;z 2Bsuch thatxyz; xy andy'z. Then there is u2B
such thatxuz; x'u anduz.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.9 Boolean Algebras Are Always One-Based 125
Proof: SetI=id() andJ=id('). Set also
j=x0^y= (x0^y)_(x^y0) =x+y
and note that j2Iby Corollary 9.1, while (4) implies x_j=x_y=y.
Similarly, setting k=y0^zwe getk2Jandy_k=z. Now take u=x_k.
Thenxux_z=zandu_k=x_k, hencex'u. Besides, jyz,
hence
z_j=z=y_k=x_j_k=u_j ;
thereforeuz. 2
Theorem 4.9.2. Every two congruences of a Boolean algebra permute.
Proof: Let;'2Con(B ). Sinceand'can be interchanged, it suces
to prove that a(')b=)a(')b. Suppose a(')b. Thenac and
c'b for somec2B. We are going to nd d2Bsuch thata'd anddb.
Sincea(a_c) andc'(b_c), whence (a _c)'(a_b_c), we can apply
Lemma 9.3 and nd an element esuch thataea_b_c; a'e and
e(a_b_c).
Sinceb'(b_c) andb(a_b), we apply again Lemma 9.3 and obtain
an element fsuch thatbfa_b_c; bf andf'(a_b_c).
Now taked=e^f. But (e ^f)'(e^(a_b_c)), that is, d'e, and
sincea'e, it follows that a'd. Also, (f ^e)(f^(a_b_c)), that is, df,
and sincebf, it follows that db. 2
Remark 4.9.1. A much shorter proof of Theorem 9.2 follows from Corol-
lary 9.1 and the fact that the congruences of every group permute.
If the lattice Con(A) of an algebra Ais distributive, then one says that
Aiscongruence distributive, and if every two congruences of Apermute,
thenAis said to be congruence permutable. If all the members of a variety
have a certain common property, then one says that the variety itself has
that property. In particular by an arithmetical variety is meant a variety
which is both congruence distributive and congruence permutable.
Now from Theorems 9.1 and 9.2 we obtain
Corollary 4.9.3. The variety of Boolean algebras is arithmetical.
Pixley [1963] characterized congruence distributive and congruence per-
mutable varieties by two ternary terms. Gould and Gr atzer [1967] com-
bined them into a single ternary term characterizing arithmetical varieties.
Then Pixley [1971] independently rediscovered the latter result. This is
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
126 4. Boolean Algebras
Theorem II.12.5 in Burris and Sankappanavar [1981], which says that a
variety is arithmetical if and only if it admits a ternary polynomial psat-
isfying the following identities:
GGP p(x;y;y ) =x; p(x;y;x) = x; p(x;x;z ) =z;
to the e ect that identities GGP hold in every algebra of that variety. In
view of this, we refer to the above identity as GGP and to a ternary term
satisfying it as a Gould-Gr atzer-Pixley term or GGP term for short. It is
important to note that being a GGP term is not an intrinsic property, but
a property relative to a variety.
Proposition 4.9.4. The unique GGPterm of a Boolean algebra is
(5) p(x;y;z ) = (x ^y0)_(y0^z)_(x^z):
Proof: In view of L owenheim's Veri cation Theorem (see e.g. Rudeanu
[1974], Theorem 2.13) an identity holds in a Boolean algebra if and only if
it is veri ed for all the values 0,1 given to the variables. Hence identities
GGP are equivalent to the following conditions:
(6.1) p(0;0;0) =p(0;1;1) = 0; p(1; 0;0) =p(1;1;1) = 1;
(6.2) p(0;0;0) =p(0;1;0) = 0; p(1; 0;1) =p(1;1;1) = 1;
(6.3) p(0;0;0) =p(1;1;0) = 0; p(0; 0;1) =p(1;1;1) = 1:
On the other hand, every Boolean polynomial satis es the interpolation
formula, which for 3 variables reads
(7) p(x;y;z ) =Wfp( ; ;
)^x ^y ^z
j ; ;
2 f0; 1gg;
where we have set x0=x0andx1=x; (see e.g. Rudeanu (op.cit), Theorem
1.60).
We see that conditions (6) are consistent. Therefore, in view of (7),
they yield the unique GGP term
p(x;y;z ) = (x ^y^z)_(x^y0^z)_(x^y0^z0)_(x0^y0^z);
which, after obvious simpli cations which use the idempotency of _, reduces
to (5). 2
Note that Proposition 9.4 provides a very short proof of Corollary 9.3.
The relevance of the above results to the axiomatics of Boolean algebras
follows from Theorem 9.3 below, known since 1967 and due to Baker, Tarski,
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.9 Boolean Algebras Are Always One-Based 127
McKenzie, Gr atzer, Quackenbush and Padmanabhan. We follow here the
proof given by Padmanabhan and Quackenbush [1973].
Theorem 4.9.3. Every nitely based arithmetical variety is one-based.
Proof: We are going to use the GGP characterization of arithmetical
varieties.
First we remark that any identity f=gis equivalent to p(u;f;g ) =u,
whereuis a variable not occurring in forg. Forp(u;f;f ) =uand con-
versely, ifp(u;f;g ) =uholds, then we take u:=fand obtainp(f;f;g ) =f;
butp(f;f;g ) =gby (4), hence f=g.
Now let us prove that every two identities can be combined into a single
identity. First we use the remark above and express the given identities as
absorption identities, say f(y;y1;:::;y n) =yandg(z;z1;:::;z m) =z, and
we claim that the system f=y; g=zis equivalent to
(8) p(x;f;y ) =p(x;g;z ):
For the system implies
p(x;f;y ) =p(x;y;y ) =x=p(x;z;z ) =p(x;g;z ):
Conversely, (8) implies y=p(f;f;y ) =p(f;g;f ) =fandg=p(g;f;f ) =
p(g;g;z ) =z.
A repeated application of the above transformations reduces any nite
system of identities to a single absorption identity, say f(z;z1;:::;z n) =z.
It remains to prove that the system consisting of the three axioms GGP
andf=zis equivalent to the single identity
(9) p(p(u;u;x);p(f;y;z );z) =x:
Clearly the system implies
p(p(u;u;x);p(f;y;z );z) =p(x;p(z;y;z );z) =p(x;z;z ) =x:
Conversely, suppose (9) holds. De ne the function f1(x;z1;:::;z n) =
f(x;
z1;:::;z n). Then (9) implies
p(p(f 1;f1;x);p(f 1;x;x);x) = x;
which shows that xcan be written in the form x=p(u;u;x), therefore (9)
implies
(10) p(x;p(f;y;z );z) =x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
128 4. Boolean Algebras
whence
(11) p(p(u;u;x);p(f;y;z );z) =p(u;u;x):
It follows from (9) and (11) that
(12) p(u;u;x) = x:
By applying in turn (12) and (10) we obtain
(13) z=p(p(f;y;z );p(f;y;z );z) =p(f;y;z );
therefore (10) becomes
(14) p(x;z;z ) =x:
By applying in turn (13) and (14) we get
z=p(f;z;z ) =f
so that (13) becomes z=p(z;y;z ), completing the proof of GGP. 2
Corollary 4.9.4. Every nitely based variety which admits a GGP-term
is one-based.
Proof: By Theorem 9.3 and Pixley's theorem. 2
Theorem 4.9.4. The variety of Boolean algebras is one-based, whatever
be the type.
Proof: Under term equivalence (i.e. de nitional equivalence), the poly-
nomials remain the same, and hence it is easy to see that the congruences
are the same, so that the variety remains arithmetical. Since the nite
basis property remains invariant under term-equivalence, the variety of all
Boolean algebras { of any nite type { is obviously nitely based and hence
is one-based by Theorem 9.3 and Corollary 9.3. 2
4.10. Orthomodular Lattices
Orthomodular lattices generalize Boolean algebras. They have arisen in the
study of quantum logic, that is, the logic which supports quantum mechan-
ics and which does not conform to classical logic. As noted by Birkho
and von Neumann [1936], the calculus of propositions in quantum logic
\is formally indistinguishable from the calculus of linear subspaces [of a
Hilbert space] with respect to set products, linear sums and orthogonal
complements" in the roles of and, or andnot, respectively. This has led to
the study of the closed subspaces of a Hilbert space, which form an ortho-
modular lattice in contemporary terminology. As oftenhappens in algebraic
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.10 Orthomodular Lattices 129
logic, the study of orthomodular lattices has tremendously developed, both
for their interest in logic and for their own sake; see Kalmbach [1983].
This section is devoted to the axiomatics of orthonormal lattices and of
the more general class of ortholattices. After the necessary de nitions, we
present a few systems of axioms and several metatheorems.
Anortholattice is an algebra (L; _;^;0;0;1) such that the reduct
(L;_;^;0;1) is a bounded lattice and the unary operation0satis es the
De Morgan laws, the law of double negation x00=xand the complementa-
tion lawsx_x0= 1 andx^x0= 0.
Note that it suces to postulate only one of the De Morgan laws. For
instance, if (x _y)0=x0^y0holds, then
(x^y)0= (x00^y00)0= (x0_y0)00=x0_y0:
Roughly speaking, one could say that an ortholattice is a Boolean alge-
bra without distributivity. The inclusion of Boolean algebras within ortho-
lattices can be speci ed as follows.
Proposition 4.10.1. The following conditions are equivalent for an ortho-
lattice L:
(i)L is a Boolean algebra ;
(ii)L is distributive ;
(iii)L is uniquely complemented .
Proof: (i)()(ii): By the de nition of ortholattices.
(ii)=)(iii): By Proposition 1.1.
(iii)=)(i): By Theorem 8.1, condition 3. 2
Anorthomodular lattice is an ortholattice which satis es the orthomod-
ular law
OMxy=)x_(x0^y) =y:
Clearly OM implies that every modular ortholattice is an orthomodular
lattice. But, despite the terminology, an orthomodular lattice need not be a
modular lattice. Kalmbach (op.cit.), Ch.1, Exercise 20 gives an example of
an orthomodular lattice which is not modular and two examples of modular
ortholattices (hence orthomodular lattices) that are not distributive. One
of them is the \Chinese lantern", i.e., the six-element lattice having four
pairwise incomparable elements a;b;c;d witha0=b; b0=a; c0=d; d0=
c;00= 1; 10= 0.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
130 4. Boolean Algebras
As for the equational characterization (2.1.2) of modularity (2.1.1), it
is easy that OM is equivalent to
OM1 x_(x0^(x_y)) =x_y;
and also to
OM2 (x ^y)_((x^y)0^y) =y;
therefore the orthomodular lattices form a subvariety of the equational class
of ortholattices. Note also that the concepts of ortholattice and orthomod-
ular lattice are self-dual. To see this just write OM1 or OM2 for x0andy0
and take complements of both sides. Likewise, any variety of ortholattices
is self-dual.
A few short independent systems of axioms for the above algebras have
been given.
Thus, ortholattices were de ned by Soboci nski [1975b] via the system
fO1; O2;O3;O4g, where
O1a_b=b_a;
O2a^(a_b) =a;
O3a_(b^b0) =a;
O4 (a _b)_c= ((c0^b0)0^a0)0;
by the simpler system fO2; O3;O5gdue to Beran [1976], where
O5 (a _b)_c= (c0^b0)_a;
and also by the system fO6; O7gdue to Soboci nski [1979], where
O6 (b ^(c^a))_a=a;
O7 ((a ^(b^(c_c)))_d)_e= ((((g ^g0)_(c0^f0)0)^(a^b))_e)_
((h_d)^d):
Orthomodular lattices are de ned by systems fO2; O3;OM3gand
fO6; OM4g, given by Soboci nski [1976b] and [1979], respectively, where
OM3 a_((a_((b_c)_d))^a0) = ((d0^c0)0_b)_a;
OM4 ((((a0^(a_((b_c)_d)))_a)^(m^n))_g)_e
= ((n^(((h0^h)_(((d0^c0)0_b)_a))^m))_e)_((f_g)^g):
Modular ortholattices are de ned by systems fMO1; MO2; MO3gand
fO6; MO4g, given by Soboci nski [1976a] and [1979], respectively , where
MO1 ((a ^b)_(a^c))_(d^d0) = ((c ^a)_b)^a;
MO2 (b _a)^a=a;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.10 Orthomodular Lattices 131
MO3 (a _b)_c=a_(b0^c0)0;
MO4 (((a ^(b_c))^(m^(g_n)))_d)_e
= ((((h ^h0)_(n0^g0)0)^((a^((b^(a_c))_c))^m))_e)_
((f_d)^d):
We prove below two of these results. However for the independence part
the reader is referred to the original papers.
Proposition 4.10.2. ((Beran [1976]) The system fO2; O3;O5gcharacter-
izes ortholattices.
Proof: It follows from O2 with b:=b^b0and O3, that a^a=a.
Therefore, taking b:=c:=c^c0in O5 we obtain, via O3,
(1) a= (c^c0)00_a:
Further, taking a:=c^c0in (1) and using again O3, we get c^c0= (c^c0)00,
which transforms (1) into
(2) a= (c^c0)_a:
Now O3 and (2) imply (c ^c0)_(b^b0) =c^c0andb^b0= (c^c0)_(b^b0),
respectively, hence c^c0=b^b0. Therefore this element is a constant, say
0, so that O3 and (2) read
(3) a_0 =a= 0_a:
Moreover, we have seen that 0 = 000. Therefore, setting 00= 1, we have
also 0 = 10.
Takingc:= 0 in O5 we obtain
(4) a_b= (1^b0)0_a;
which fora:= 0 yields b= (1^b0)0and this transforms (4) into
(5) a_b=b_a:
Besides, we have also
(6) b_c= (c0^b0)0
by O5 with a:= 0, hence a0_a= (a0^a00)0= 00= 1, so that the two
complementation laws hold.
Furthermore, O2 implies a0=a0^(a0_a) =a0^00, whence using once
again (6) we get a00= (a0^00)0= 0_a=aand we have thus proved
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
132 4. Boolean Algebras
(7) a00=a:
Now from (7), (6) and (5) we infer
b^c= (b00^c00)00= (c0_b0)0= (b0_c0)0= (c00^b00)00=c^b;
while from O5, (6) and (5) we infer
(a_b)_c= (c0^b0)0_a= (b_c)_a=a_(b_c):
At this point it is routine to use the De Morgan law (6) together with
(7) in order to prove the associativity of ^and the duals of O2 and (3).
2
Lemmas 10.1, 10.2 and Proposition 10.3 below are due to Soboci nski
[1976a].
Lemma 4.10.1. MO1, MO2 andMO3imply O3.
Proof: It follows from MO1 and MO2 that
(8) ((a^a)_(a^c))_(d^d0) = ((a ^a)_a)^a=a;
then from (8) and MO2 we get
(9) a^(d^d0) = (((a ^a)_(a^c))_(d^d0))^(d^d0) =d^d0;
whence (8), (9) and MO3 imply
a= ((a^a)_(a^(d^d0)))_(d^d0) = ((a ^a)_(d^d0))_(d^d0)
= ((a^a)_(d^d0))_(d^d0) = (a ^a)_((d^d0)0^(d^d0)0)0;
therefore setting
= ((d^d0)0^(d^d0)0)0;
we havea= (a^a)_. This implies a^= ((a^a)_)^=by MO2,
hence using (8) we obtain
a= ((a^a)_(a^))_(d^d0) = ((a ^a)_)_(d^d0) =a_(d^d0):
2
Lemma 4.10.2. MO1, MO2 andMO3imply
(10) (a^b)_(a^c) = ((c ^a)_b)^a;
(11) a= (a^a)_(a^c);
and the idempotency laws.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.10 Orthomodular Lattices 133
Proof: Identity O3 holds by Lemma 10.1, therefore MO1 reduces to (10).
From MO2 with c:=c^awe obtain (11). Now (11) and (10) imply
(12) a^a= ((a^a)_(a^a))^a= ((a^(a^a))_(a^a);
while from MO2 and (11) we infer
(13) a^a= ((a^a)_(a^a))^(a^a) =a^(a^a);
so that (11), (13) and (12) imply
a= (a^a)_(a^a) = (a ^((a^a))_(a^a) =a^a;
which implies further a= (a^a)_(a^a) =a_a, via (11). 2
Proposition 4.10.3. The system fMO1; MO2; MO3gcharacterizes mod-
ular ortholattices.
Proof: First we note that it suces to prove O2 and the commutativity
laws. For commutativity reduces MO3 to O5, while O3 holds by Lemma
10.1, therefore the algebra will be an ortholattice by Proposition 10.2. Be-
sides, Lemma 10.2 ensures that (10) holds, and this identity will coincide
with the modularity condition (2.1.20) due to commutativity.
We are going to use the conclusions of Lemma 10.2. First we prove O2.
It follows from MO2 and (11) that
(14) a^c= ((a^a)_(a^c))^(a^c) =a^(a^c);
while (10) implies
(15) (a^b)_a= (a^b)_(a^a) = ((a ^a)_b)^a= (a_b)^a;
and using (11), (10), (14) and (15) we get
(16)a=a^a= ((a^a)_(a^c))^a= (a^(a^c))_(a^a)
= (a^c)_a= (a_c)^a;
while using (16) twice, then (10), we obtain O2:
a= (a_(a_b))^a= (((a _b)^a)_(a_b))^a
= (a^(a_b))_(a^(a_b)) =a^(a_b):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
134 4. Boolean Algebras
Finally the desired commutativities are obtained as follows. From MO2
and (16), then (10) and O2 we get
a_b= ((b_a)^a)_((b_a)^b) = ((b ^(b_a))_a)^(b_a)
= (b_a)^(b_a) =b_a;
while (10), (15) and (16) imply
a^b= (a^b)_(a^b) = ((b ^a)_b)^a= ((b_a)^b)^a=b_a:
2
In the second part of this section we prove several metatheorems in
which GGP terms and a generalization of them, Mal'cev terms, play a
crucial role.
Remark 4.10.1. In every orthomodular lattice
p(x;y;z ) = (x ^z)_((y^z)0^z)_((x^y)0^x)
is obviously a GGP term
Theorem 4.10.1. Every nitely based variety of orthomodular lattices is
one-based.
Proof: By Corollary 9.4 and Remark 10.1. 2
Corollary 4.10.1. The variety of orthomodular lattices is one-based.
Furthermore, let O 8be the ortholattice obtained by adjoining 0 and 1
to the two incomparable chains a < b < c andc0< b0< a0. By removing
candc0one obtains an ortholattice known as O 6. Clearly O 6is not an
orthomodular lattice (OM fails for x:=a; y :=b) and in fact it is known
that an ortholattice is orthomodular if and only if it does not include O 6as a
subalgebra; see e.g. Kalmbach [1983], Ch.1, Theorem 2. We will refer to this
characterization of orthomodularity as the forbidden-subalgebra theorem.
Lemmas 10.3, 10.4, Theorems 10.2, 10.3 and Corollaries 10.2, 10.3 below
are due to D.Kelly and Padmanabhan [2005].
Lemma 4.10.3. The ortholattice O8is not congruence-permutable.
Proof: The principal congruences (a;b) and(b;c) satisfy a(a;b)b &
b(a;b)c, but (c;a) 62(a;b)(b;c). 2
Theorem 4.10.2. A variety of ortholattices is congruence-permutable if
and only if it is a variety of orthomodular lattices.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
4.10 Orthomodular Lattices 135
Proof: Every two congruences of an orthomodular lattice permute; cf.
Kalmbach [1983], Ch.2, Exercise 4. Conversely, let Vbe a variety of or-
tholattices that is congruence-permutable. Since O 8is a subalgebra of the
square of O 6,Vdoes not contain O 6by Lemma 10.3, therefore every algebra
inVis orthomodular by the forbidden-subalgebra theorem. 2
To obtain further results, we recall a theorem due to Mal'cev, which
says that a variety Vis congruence-permutable if and only if there is a
termp(x;y;z ) such that the following identities hold in every V-algebra:
MV1 p(x;y;y ) =x;
MV2 p(y;y;x) = x:
Such a polynomial pis called a Mal'cev term; see e.g. Gr atzer [1979],
Theorem 26.4, or Burris and Sankappanavar [1981], Ch.2, Theorem 12.2.
Note that every GGP term (cf. x9) is a Mal'cev term and the property of
being a Mal'cev term is also relative to a variety.
Lemma 4.10.4. Let p be a Mal'cev term for orthomodular lattices. Then
orthomodular lattices are characterized by MV1orMV2among ortholattices
according as MV1orMV2fails in O6.
Proof: Since O 6is not orthomodular, the variety generated by it is not
congruence-permutable, by Theorem 10.2. It follows by the theorem of
Mal'cev that O 6cannot satisfy both MV1 and MV2. If MV1 fails in O 6
then any ortholattice that satis es MV1 has not O 6as a subalgebra, hence
it is orthomodular by the forbidden-subalgebra theorem. The other case is
treated similarly. 2
Theorem 4.10.3. An ortholattice is orthomodular if and only if it has a
GGP-term.
Proof: The \only if" part is Remark 10.1. Conversely, an ortholattice
having a GGP term is orthomodular by Lemma 10.4. 2
Corollary 4.10.2. If p is a ternary term symmetric in x an z and such
thatMV1holds in orthomodular lattices, then condition MV1characterizes
orthomodular lattices among ortholattices.
Proof: Conditions MV1 and MV2 are now equivalent, therefore pis a
Mal'cev term and the desired conclusion follows by Lemma 10.4. 2
Corollary 10.2 is a generator of several relative characterizations of or-
thomodular lattices. Here is an example:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
136 4. Boolean Algebras
Corollary 4.10.3. Identity
(x^y)_((x^y)0^x) =x
characterizes orthomodular lattices among ortholattices.
Proof: By Corollary 10.2 and Remark 10.1. 2
The paper by D. Kelly and Padmanabhan [2003] provides several results
in this line. The main one is that for every n2, every nitely based variety
of orthomodular lattices has an independent self-dual n-basis.
The following companion of Theorem 10.3 is also due to D.Kelly and
Padmanabhan [2007]:
Theorem 4.10.4. An orthomodular lattice is a Boolean algebra if and only
if it has a unique GGPterm.
Proof: The \if" part is Proposition 9.4. To prove the \only if" part,
note that Remark 10.1 and its dual provide two GGP terms. According
to the hypothesis, they must coincide, so that identity B_^
2holds, which
characterizes Boolean algebras by Proposition 2.4. 2
Several GGP terms for orthomodular lattices have been pointed out in
the literature. In a Boolean algebra they coincide with the unique term
described in Proposition 9.4. Alternatively, this can be easily checked by
direct computation in Boolean algebra.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Chapter 5
Further Topics and Open Problems
In this short chapter we present, in a rather informal and sketchy way, a
few lines of research related to the material of this book and leading to
several open problems.
I. Tarski-type theorems on independent equational bases
The topic of this monograph belongs to what is widely known as equa-
tional logic and equational theories of algebras. We may safely say that
Alfred Tarski School at the University of California, Berkeley (during the
late 1960's) was, in a sense, the Mecca for equational logic in modern times.
LetKbe a nitely-based equational theory of algebras. Following Tarski
[1968], let r(K) denote the equational spectrum of K, that is, the set of
cardinalities of independent equational bases of K. Tarski has shown that
ifKsatis es an equation of the form f=x, wherefhas at least two occur-
rences of the variable x, then r(K) is an unbounded interval. This is what
we call Tarski's Unbounded Theorem (TUT). McNulty proved a stronger
version of TUT in 1976. For such varieties, Tarski has further shown that if
i;jbelong to the set r(K), then every integer between iandjalso belongs
tor(K). This we call Tarski's Interpolation Theorem (TIT).
For a variety admitting a duality , let rsd(K) be the set of all natural
numbersnsuch that Khas an independent self-dual basis withnidentities.
First we give a rather simple example of a nitely-based self-dual variety
which demonstrates the failure of the self-dual analogue of TIT. Indeed, if
Kis the variety of all algebras of type (2,2) with two semilattice operations,
then it is easy to show that 2, 4 belong to rsd(K). However, it has been
shown (D. Kelly and Padmanabhan [2004]) that Kcannot be de ned by any
independent self-dual set of three identities. Thus, the self-dual analogue of
Tarski's Interpolation Theorem is not true if we insist that the equational
basis in question enjoys some additional syntactic property (e.g., being self-
dual).
137
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
138 5. Further Topics and Open Problems
However, the self-dual analogues of TIT and TUT are valid for all \nice"
lattice varieties K(including distributive lattices, modular lattices, Boolean
algebras, ortholattices, OML's and any nitely-based self-dual variety of
lattices). In fact, all these self-dual equational theories have independent
self-dual bases for all n2;3 or 4. For example, in D. Kelly and Padman-
abhan [2004] it has been shown that
rsd(K) = [1;! ) ifKis the self dual variety of all one element latices;
= [2;! ) ifKis the self dual variety of all lattices;
= [3;! ) ifKsatis es a self dual identity f=df;
= [4;! ) otherwise:
Tarski's original 1975 proof (without the further syntactic restriction that
the equational bases be self-dual) uses the concept of closure system and is
existential in nature. In sharp contrast, D.Kelly and Padmanabhan exhibit
the actual self-dual bases by means of a natural blow-up process based on
elementary number-theoretic models. Thus, these independent equational
bases may be construed as the rst constructive proof of Tarski's original
theorems as well.
Open problems
1. Let Kbe the variety of all algebras of type (2,2) with two semilattice
operations. As mentioned above, 2 and 4 belong to rsd(K), but not 3.
Does 5 belong to rsd(K) ? In other words, is it possible to de ne the
semilattice properties of two operations by an independent self-dual set of
ve identities ?
2. Characterize the set rsd(K).
3. Is there an example of a variety Kof algebras admitting a duality
but having no independent self-dual bases ?
II. Huntington varieties of lattices
Unlike the classical algebraic theories like groups and rings, substruc-
tures play a rather unique role in the equtional theory of lattices. Just recall
the famous 1897 theorem of Dedekind that a lattice is modular if and only if
it contains no pentagon (Theorem 2.1.1). As early as 1943, Lowig gave the
rst example of a \forbidden" non-self-dual lattice, which was published
in the Annals of Mathematics (we don't see any research paper in lattice
theory published nowadays in such main-stream journals, do we?). That
subdirectly irreducible lattice has nine elements and is, in fact, one of the
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
5. Further Topics and Open Problems 139
covers of N 5. But its publication was way ahead of its time and did not
catch the atention of many lattice theorists. The lattice-theory world had
to wait for the landmark discoveries of bounded homomorphisms and split-
ting lattices by McKenzie (see Ch.2 in Freese, Jezek and Nation [1995])
and the Magic Lemma of J onsson along with the J onsson terms to fully
appreciate the interplay between equational theory of lattices and internal
algebraic structure of lattices. Recall that Lis asplitting lattice if there is
a lattice identity f=g(theconjugate identity of the lattice L) such that
a latticeKsatis esf=gif and only if Lis not in the variety generated
by the lattice K. McKenzie characterized splitting lattices as those nite
subdirectly irreducible lattices which are bounded epimorphic images of a
free lattice. These lattices are semidistributive (i.e., they satisfy SD_and
SD^; see Theorem 4.8.3). Moreover, every nite splitting lattice satis es
an identity which formally implies these semidistributive implications. It
was this aspect of lattice theory that we exploited in Section 8 of Chapter
4 to discover in nitely many Huntington varieties of lattices, i.e., lattice
varieties in which every uniquely complemented lattice is distributive.
Open problems
4. We conjecture there exists no largest Huntington variety.
5. We conjecture that if AandBare Huntington varieties, then so
isA_B. In support of this conjecture we have shown that M_N5is
Huntington; cf. Corollary 8.8.1.
Note that the analogue of Problem 5 for in nite joins has a negative
answer. For, using the celebrated J onsson Lemma, one can show that every
splitting lattice generates a non-modular Huntington variety. It follows
that the in nite join of all these Huntington varieties is the variety Lof all
lattices, because in fact, according to a well-known result of Alan Day, the
class of all splitting lattices generates L. But trivially the variety Lis not
Huntington. As a matter of fact, Lis a Dilworth variety; cf. Problem 9.
6. Call a lattice variety Kstrictly nonmodular ifKdoes not contain
M3. We conjecture that every strictly nonmodular variety is Huntington.
7. Prove or disprove that the variety generated by a nite lattice is
Huntington.
8. Prove or disprove that if a nite lattice Lgenerates a Huntington
variety, then so does L[d] obtained by Day's doubling construction. Here
L[d] is obtained by \doubling the element d"; see Day [1977] and Jipsen
and Rose [1992], page 88, Lemma 4.11.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
140 5. Further Topics and Open Problems
Lemma 4.11 proves that if Lis a nite distributive lattice, then L[d] is
almost distributive { in particular, it is semidistributive and hence by the
results mentioned in Chapter 4, the variety generated by L[d] is Huntington.
Our problem asks whether this is always the case, if Lis not necessarily
distributive.
9. A lattice variety Kis called a Dilworth variety if every lattice in
Kcan be embedded in some uniquely complemented lattice in K. For
instance, the variety of all lattices is Dilworth. Adam and Sichler proved
that everyp modular lattice can be embedded in a uniquely complemented
p modular lattice. Is there an example of a lattice variety that is neither
Huntington nor Dilworth ?
10. Project. Carry out a similar program for the bi-complemented
lattices of Chen and Gr atzer [1969]. For example, characterize semi-
distributive complemented lattices in which every element has precisely
two complements.
11. Day, Nation and Tschantz [1989] have introduced the implication
of semidistributivity: x^y=x^z&z_x=z_y=)xy. We conjecture
that if a lattice variety satis es this implication, then it is a Huntington
variety.
III. Binary reducts of Boolean algebras
In Chapter 4 we discussed several avatars of Boolean algebras and
proved that each one of them can be de ned by a single axiom. This
was an easy consequence of the fact that the variety of Boolean algebras
is arithmetical. However, most of the reducts of Boolean algebras do not
have the arithmetical property. In theory there are sixteen binary reducts
of Boolean algebras, each one corresponding to a binary term { an element
of the free Boolean algebra on two generators. By symmetry and duality
there are just eight essentially distinct binary reducts: (B;f ), where
f(x;y)2 f0;x;x0;x^y;x^y0;x0_y;x0^y0;(x^y0)_(x0^y)g:
Open problems
12. Project. Find minimal equational bases for the varieties generated
by (B;f ).
The following minimal equational bases are already known:
forf(x;y) =x(left semigroups): f(x;y) =x;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
5. Further Topics and Open Problems 141
forf(x;y) =x^y(semilattices): two identities; cf. Chapter 1, x1 (these
systems are minimal by Theorem 1.3.1);
forf(x;y) =x0^y0(She er stroke): single identity; cf. Chapter 4, Bj
87;
forf(x;y) =x0_y(implication algebras): two identities; see Theorems
5.3.1 and 5.3.2 below.
13. We conjecture there is no single identity de ning Boolean algebras
shorter than Bj
87among all possible single identities (i.e., not only among
those expressed in terms of the She er stroke).
The implicational fragment of the two-element Boolean algebra is the
class of algebras of type (2) having a single binary operation !with the
interpretation that x!y=x0_y. Abbott [1967] rst de ned these
implication algebras by the following three identities:
(1) (x!y)!x=x;
(2) (x!y)!y= (y!x)!x;
(3) x!(y!z) =y!(x!z):
It is natural to look for a minimal equational basis of implication alge-
bras. This problem is solved by Theorems 5.3.1 and 5.3.2 below, due to
Gareau and Padmanabhan (unpublished). We are going to use the well-
known fact that the relation xy()x!y= 1 makes an implication
algebra into a poset with greatest element 1, where 1 = z!zfor anyz.
This implies the identity x!1 = 1. Note also that implication algebras
with 0 coincide with Boolean algebras.
Lemma 5.3.1 Letf(x1;:::;x n)be a non-constant term function of an im-
plication algebra L. If L satis es an identity of the form f(x1;:::;x n) =xi,
where 1in, thenxiis the last variable which occurs in f.
Comment: In particular Lmay be a Boolean algebra.
Proof: This follows from the identity xnf(x1;:::;x n), wherexn
stands for the last variable which appears in f. We prove the identity by
induction on the length of the expression of f. Forf:=x!ywe note
that (3) implies
y!(x!y) =x!(y!y) =x!1 = 1;
i.e.,yx!y. The inductive step follows from xnf2f1!f2=f.
2
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
142 5. Further Topics and Open Problems
Theorem 5.3.1 Implication algebras are not one-based.
Proof: Supposef(x1;:::;x n) =g(x1;:::;x n) is a single axiom for im-
plication algebras. If none of the term functions f;greduces to a single
variable, then the constant algebra x!y= 1 would be a model for that
axiom; but this algebra does not satisfy axiom (1). For the same reason
the axiom cannot be of the form f(x1;:::;x n) = 1 either. Therefore the
single axiom is of the form f(x1;:::;x n) =xi, where 1 in.
Now it follows by Lemma 5.3.1 that the single identity is f(x1;:::;x n) =
xn. Then the right projection x!y=yis a model for the single axiom,
but it does not ful l axiom (2). 2
Theorem 5.3.1 can be extended to orthomodular lattices in the following
way. As shown by Kalmbach [1983], Ch.4, x15, Theorem 3, in every ortho-
modular lattice there are exactly ve term functions !jwith the property
xy()x!jy= 1:
If the OML is a Boolean algebra, all of them reduce to x!y=x0_y.
Corollary 5.3.1 The equational theory of (L;!j) (1jn)has no
single axiom.
Proof: Suppose one of the ve implications !jis characterized by a
single axiom. As in the proof of Theorem 5.3.1, the axiom must be of
the formf(x1;:::;x n) =xi. But this axiom is ful lled by the implication
x!y=x0_yof a Boolean algebra. So it follows by Lemma 5.3.1 that
i=nand again the right projection is a model of the axiom but fails to
satisfy the commutativity of the disjunction operation of the OML, which
is expressed by an identity '(x;y ) = (x;y), where the last variables in '
and arexandy. 2
Theorem 5.3.2 The identities (1)and
(4) x!(y!((z!u)!u)) =y!(x!((u!z)!z))
form a basis for the equational theory of implication algebras.
Proof: Note rst that axioms (3) and (2) imply (4). The following proof
of the converse implication is based on an automated proof provided by
Prover9.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
5. Further Topics and Open Problems 143
Settingu:=zin (4) and taking into account (1) we obtain (3). So it
remains to prove (2).
By applying (1) twice we obtain
(5) x!(x!y) = ((x !y)!x)!(x!y) =x!y:
By applying (4) with
x:= (((y !x)!x)!z); y:=u; z :=x; u :=y;
then (1) with x:= (y!x)!x; y :=z, we obtain
(6)(((y!x)!x)!z)!(u!((x!y)!y))
=u!((((y!x)!x)!z)!((y!x)!x)
=u!((y!x)!x):
Further, by using in turn (3), (1), (6) with u:= ((x !y)!y)!u,
and (3), we obtain
(x!y)!((((y!x)!x)!z)!y)
= (((y !x)!x)!z)!((x!y)!y)
= (((y !x)!z)!((((x!y)!u)!((x!y)!y))
= (((x !y)!y)!u)!((y!x)!x)
= (y!x)!((((x!y)!y)!u)!x);
hence
(7)(x!y)!((((y!x)!x)!z)!y)
= (y!x)!((((x!y)!y)!u)!x):
On the other hand, using the abbreviation ((y !x)!x)!z=uand
applying in turn (3), (5), (6) and (1), we obtain
(x!y)!(u!y) =u!((x!y)!y) =u!(u!((x!y)!y))
= (((y !x)!x)!z)!((y!x)!x) = (y !x)!x;
that is,
(8) (x!y)!((((y!x)!x)!z)!y) = (y !x)!x:
By interchanging xandywe get
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
144 5. Further Topics and Open Problems
(80) (y!x)!((((x!y)!y)!z)!x) = (x !y)!y;
but the left-hand sides of (8) and (80) coincide by (7), therefore (y !x)!
x= (x!y)!y, which is (2). 2
Historic remark The fact that this equational theory is 2-based was rst
proved by Meredith and Prior [1968], who gave a system consisting of
equations (1) and
(9) (x!y)!(z!y) = (y !x)!(z!x):
Corollary 5.3.2 The complete equational spectrum for this variety is [2;!),
i.e., for all n2there exists an independent n-basis for implication alge-
bras.
Proof: Theorem 8 in the paper by Tarski [1968] says that if is an
equational theory, is a term, xis a variable with at least two occur-
rences in such that the equation =xbelongs to and 0 <n2 r(),
then [n;! ) r(). In view of the above theorems, we can apply Tarski's
theorem with := (x!y)!xandn:= 2. 2
The above corollary can be further extended. Consider, for instance,
the Boolean subtraction functionx y=x^y0, which is the dual of
y!xand can be referred to as the mirror image orskew dual of the
implication. Kalman [1960] characterized the equational theory of algebras
(L; ) satisfying all the identities true for the Boolean function x^y0by
the system
(10) x (y x) =x;
(11) x (x y) =y (y x);
(12) (x y) z= (x z) (y z):
(Incidentally, (10) and (11) are the mirror images of (1) and (2), respec-
tively.) Kalman called these algebras by the name of
ocks.
Corollary 5.3.3 The complete equational spectrum for the Kalman variety
of
ocks is [2;!).
Proof: From Corollary 5.3.2 by skew duality. 2
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
5. Further Topics and Open Problems 145
IV. Frink-type theorems for varieties of complemented lattices
Quoting G. Birkho , O. Frink gave a remarkably brief and elegant proof
of the well-known theorem that every Boolean algebra is isomorphic with
an algebra of sets. The novelty of the approach was that it was based
on a simple bi-implication capturing the meaning of the order relation in
Boolean algebras: x^y=x()x^y0= 0. The profound direction is,
of course, the suciency of x^y0= 0 forx^y=xand this characterizes
Boolean algebras among all semilattices with an extra unary operation.
More generally, let Kbe a variety of complemented lattices. A Frink-
type theorem for Kis a statement of the form: a semilattice with a unary
operation belongs to Ki it satis es the implication
f(x;y) =g(x;y) =)x^y=x;
wherefandgare terms of type (2,1). Apart from generalizing the orig-
inal Frink theorem on Boolean algebras, every such result will give a new
\equational expression" for the order relation x^y=xwhich is rather
unique to the particular equational theory in question. For example, Adam
Gareau, a graduate student at the University of Manitoba, has proved such
an analogue for the class of orthomodular lattices:
(L;^;0;0) is an OML if and only if (x0^(x0^y)0)0=y=)x^y=x:
Open problems
14. Project. Discover new Frink-type theorems for well-known varieties
of complemented lattices, e.g. ortholattices, modular ortholattices, Stone
lattices and Newmann algebras.
In the case of orthomodular lattices, such a theorem would have obvious
implications (no pun intended!) to the various implications that can be
de ned in the context of the logic of quantum mechanics.
In Problems 15{22 below (L; _;^;0) is an orthomodular lattice and
f(x;y) is a binary term of L. Problems 16{20 are conjectures.
15. Find all binary reducts (L;f ) ofLwhich have a single axiom for
their equational theory.
This problem is decidable, because the free OML on two generators is
nite; cf. Kalmbach [1983], p. 218.
16. If the equational theory of the binary reduct (L;f ) is not one-based,
then it is two-based.
17. If (L;f ) is cancellative, then (L;f ) is an Abelian group.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
146 5. Further Topics and Open Problems
18. If (L;f ) is a cancellative semigroup, then (L;f ) is an Abelian group.
19. If (L;f ) is a group, then (L; _;^;0) is a Boolean algebra.
20. If (L;f ) is cancellative, then the lattice reduct (L; _;^) is distribu-
tive.
21. Characterize the variety generated by (L;x ^y0) in OML's (i.e.,
analogue of Kalman's theorem for orthomodular lattices).
22. Project. Same problems as above for the ve implications de ned
in an OML ; see Kalmbach [1983].
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix A: Some Prover9 Proofs
by W. McCune
This Appendix contains several examples of lattice theory proofs found
by the program Prover9 (version March-2007); cf. McCune [1]. For each
example, the Prover9 input le is given, followed by the proof produced by
Prover9.
Prover9 proves theorems by contradiction, and the conclusion of the
theorem is assumed to be false. For example, if the conclusion is that the
meet operation is commutative and associative, the input le contains the
following assumption (!= is negated equality, and |is disjunction).
A ^ B != B ^ A | (A ^ B) ^ C != A ^ (B ^ C).
The terms A;B;C are constants, and terms x;y;z;u;v;w are variables.
Each step of the proof contains a justi cation on the right-hand side listing
the lines from which the step was derived.
Sholander's 2-Basis for Distributive Lattices
Here is a proof of Sholander's Theorem 3.2.1.
Input le:
formulas(assumptions).
% Sholander's 2-basis for distributive lattices:
x ^ (x v y) = x # label(Absorb_A).
x ^ (y v z) = (z ^ x) v (y ^ x) # label(Sholander).
% Denial of a standard 6-basis for Lattice Theory.
A ^ B != B ^ A |
147
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
148 Appendix A: Some Prover9 Proofs
(A ^ B) ^ C != A ^ (B ^ C) |
A ^ (A v B) != A |
A v B != B v A |
(A v B) v C != A v (B v C) |
A v (A ^ B) != A.
end_of_list.
Prover9 proof in 1 second:
1 x^(x_y) =x# Absorb A [assumption]
2 x^(y_z) = (z ^x)_(y^x) # Sholander [assumption]
3 (x ^y)_(z^y) =y^(z_x) [2]
4 A^B6=B^Aj(A^B)^C6=A^(B^C)jA^(A_B)6=Aj
A_B6=B_Aj(A_B)_C6=A_(B_C)jA_(A^B)6=A
[assumption]
5 B^A6=A^Bj(A^B)^C6=A^(B^C)jB_A6=A_Bj
(A_B)_C6=A_(B_C)A_(A^B)6=A [4,1]
6 (x ^y)^(y^(z_x)) = x^y [3,1]
7 (x ^y)_(y^y) =y [3,1]
8 x_(y^(x_z)) = (x _z)^(y_x) [1,3]
9 (x ^(y_z))_y= (y_z)^(y_x) [1,3]
10 ((x ^y)_(z^y))_(u^(z_x)) = (z _x)^(u_y) [3,3]
11 x^((x_y)^(z_x)) = x [1,6,1]
12 (x ^y)^y=x^y [1,6]
13 (x ^y)^((y^(z_x))^(u_(x^y))) = x^y [6,6,6]
14 x_((x_y)^(x_y)) =x_y [1,7]
15 ((x ^y)_(z^y))_((z_x)^(z_x)) = z_x [3,7]
16 (x _y)^(x_x) =x_x [1,8]
17 (x ^(y_y))_(y_y) =y_y [8,7,1]
18 x_(y^(x_z)) = (x _z)^((x_z)^(x_y)) [12,8,9]
19 x_(x_x) =x_x [8,14,1]
20 (x ^x)_x=x [7,19,7]
21 (x ^y)_((x^x)^y) =y^x [20,3]
22 x^((x_y)^x) =x [20,11]
23 x^(((x_y)^x)^(z_x)) = x [22,6,22]
24 x_(((x_y)^x)^((x_y)^x)) = (x _y)^x [22,7]
25 (x _x)_(y^(x_x)) = (x _x)^(y_(x_z)) [16,3]
26 (x _x)^((x_y)_z) =x_x [16,3,17]
27 (x _x)^((x_x)^(y_(x_z))) = x_x [16,6,16]
28 x^((y^x)_(y^x)) = (y ^x)_(y^x) [7,16]
29 ((x ^x)_y)^x=x [7,16,7]
30 x^x=x [20,16,7,7]
31 (x _y)^x=x [29,30]
32 x_x=x [24,31,31,30,31]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix A: Some Prover9 Proofs 149
33 x^y=y^x [21,30,32]
34 ((x ^y)_(z^y))_(z_x) =z_x [15,30]
35 (x ^y)_y=y [7,30]
36 x^(x^(y_x)) = x [23,33,1]
37 x^(y^x) =y^x [28,32,32]
38 x^(x^(y_(x_z))) = x [27,32,32,32]
39 x^((x_y)_z) =x [26,32,32]
40 x_(y^x) =x^(y_(x_z)) [25,32,32,32]
41 C^(A^B)6=A^(B^C)jB_A6=A_Bj(A_B)_C
6=A_(B_C)jA_(A^B)6=A [5,33,33]
42 (x ^y)_(z^x) =x^(z_y) [33,3]
43 (x ^y)^(x^(z_y)) =y^x [33,6]
44 (x ^y)_x=x [33,35]
45 (x _(y^x))_y=y_x [10,30,30,1]
46 x^(x^y) =x^y [44,1,33]
47 (x ^y)^(x^(z_(x^y))) = x^y [44,11]
48 x^(y_(x_z)) =x [38,46]
49 x^(y_x) =x [36,46]
50 x_(y^(x_z)) = (x _z)^(x_y) [18,46]
51 x_(y^x) =x [40,48]
52 x_y=y_x [45,51]
53 x_(x^y) =x [44,52]
54 C^(A^B)6=A^(B^C)jC_(A_B)6=A_(B_C) [41,52,52,53]
55 x_(y^(z_x)) = (z _x)^(x_y) [49,3,52]
56 (x ^y)_(z^y) =y^(x_z) [52,3]
57 (x ^(y_z))_(z_y) =z_y [34,56]
58 (x ^y)^(z_y) =x^y [37,6,46,37]
59 x^((y_x)_z) =x [52,39]
60 (x ^y)^(z_x) =x^y [46,6,46,46]
61 x^(y_(z_x)) = x [52,48]
62 (x ^y)^(x_z) =x^y [53,59]
63 (x ^y)^(x^(y^(z_x))) = x^y [53,13,33]
64 (x _y)_(z^y) =x_y [58,51]
65 x^(y^(z^x)) = y^(z^x) [51,58,33]
66 x^(y^(x^z)) =y^(x^z) [53,58,33]
67 (x _y)_(y^z) =x_y [60,51]
68 x_(y^(x^z)) =x [53,64,53]
69 x^(y^(z^(x^u))) = y^(z^(x^u)) [68,58,33]
70 ((x ^y)^z)^(x^(u_y)) = (x ^y)^z [42,62]
71 (x ^(y^z))^(x^(u_y)) = (y ^z)^x [67,43]
72 (x ^y)^(x^(y^z)) =x^(y^z) [68,47,33]
73 x^(y^(z_x)) = x^y [63,72]
74 x^((y_x)^z) =x^z [33,73]
75 (x ^y)^(y^z) = (x ^y)^z [51,74]
76 (x ^y)^(x^z) = (x ^y)^z [53,74]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
150 Appendix A: Some Prover9 Proofs
77 (x ^y)^z=x^(y^z) [72,76,75]
78 x^(y^(z^(u_x))) = x^(y^z) [71,77,77,69,77]
79 x^(y^z) =z^(x^y) [70,77,77,77,78,65,77]
80 x^(y^z) =y^(x^z) [76,77,66,77]
81 C_(A_B)6=A_(B_C) [54,79,33,80]
82 (x _y)^(x_(y_z)) =x_y [48,50,33]
83 (x _y)^(x_(z_y)) =x_y [61,50,33]
84 (x _y)^(y_(x_z)) =y_x [52,82]
85 (x _y)_(z_x) = (x _y)_z [82,55,84]
86 (x _y)_(z_y) = (x _y)_z [83,55,84]
87 (x _y)_z= (z_y)_x [83,57,85,86]
88 (x _y)_z=x_(z_y) [87,52]
89 2 [88,81,52,52]
Self-dual 2-Basis for Lattice Theory
Here we prove Theorem 1.2.5.
Input le:
formulas(assumptions).
% Self-dual (independent) 2-basis for Lattice Theory
(((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y\\
# label(A).
(((x ^ y) v y) ^ (z v y)) v (u ^ ((v ^ y) v (y ^ w))) = y\\
# label(Dual_A).
% Denial of McKenzie's 4-baiss for Lattice Theory.
A v (B ^ (A ^ C)) != A |
A ^ (B v (A v C)) != A |
((B ^ A) v (A ^ C)) v A != A |
((B v A) ^ (A v C)) ^ A != A.
end_of_list.
Prover9 proof in 117 seconds:
1 (((x _y)^y)_(z^y))^(u_((v_y)^(y_w))) = y# A [assumption]
2 (((x ^y)_y)^(z_y))_(u^((v^y)_(y^w))) = y
# Dual A [assumption]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix A: Some Prover9 Proofs 151
3 A_(B^(A^C))6=AjA^(B_(A_C))6=Aj
((B^A)_(A^C))_A6=Aj((B_A)^(A_C))^A6=A
[assumption]
4 (((x _(y_((z_u)^(u_v))))^(y_((z_u)^(u_v))))_u)^(w_
((v6_(y_((z_u)^(u_v))))^((y_((z_u)^(u_v)))_v7)))
=y_((z_u)^(u_v)) [1,1]
5 (((x _(y^z))^(y^z))_(u^(y^z)))^(v_z) =y^z [1,1]
6 (((x _(y^((z^u)_(u^v))))^(y^((z
^u)_(u^v))))_(w^(y^((z^u)_(u^v)))))^
(v6_(u^((y^((z^u)_(u^v)))_v7))) = y^((z^u)_(u^v)) [2,1]
7 (((x _(y^z))^(y^z))_(u^(y^z)))^z=y^z [2,1]
8 (((x ^(y_z))_(y_z))^(u_(y_z)))_z=y_z [1,2]
9 (((x ^(y_z))_(y_z))^(u_(y_z)))_(v^z) =y_z [2,2]
10 (((x _(y_z))^(y_z))_(u^z))^(v_((w_(y_z))^((y_z)_v6)))
=y_z [1,4,1,1,1,1]
11 (((x ^y)_(z_y))^(u_(z_y)))_(v^((w^(z_y))_((z_y)^v6)))
=z_y [5,2]
12 (((x ^y)_y)^(z_y))_(u^(y^v)) =y [5,2]
13 (((x _(y^(z^((u^v)_(v^w)))))^(y^(z^((u^v)_(v^w)))))_
(v6^(y^(z^((u^v)_(v^w))))))^ _=y^(z^((u^v)_(v^w)))
[2,5]
14 (((x ^y)_(z_y))^(u_(z_y)))_y=z_y [5,8]
15 (((x ^(((y_z)^z)_(u^z)))_(((y_z)^z)_(u^z)))^(v_
(((y_z)^z)_(u^z))))_(w^z) = ((y _z)^z)_(u^z) [1,12]
16 (((x ^(y^((z^u)_(u^v))))_(y^((z^u)_(u^v))))^u)
_(w^((y^((z^u)_(u^v)))^v6)) = y^((z^u)_(u^v)) [2,12]
17 ((x ^(y^(x^z)))_(u^(y^(x^z))))^(x^z) =y^(x^z) [12,7]
18 ((x ^(y^(x^z)))_(u^(y^(x^z))))^(v_(x^z)) =y^(x^z) [12,5]
19 (((x _(y^(z^(u^v))))^(y^(z^(u^v))))_(w^(y^(z^(u^v)))))
^u=y^(z^(u^v)) [12,5]
20 (((x ^y)_(z_y))^(u_(z_y)))_(v^((z_y)^w)) =z_y [5,12]
21 (((x ^(y^(z^u)))_z)^(v_z))_(y^(z^u)) = z [12,14,12,12]
22 (((x _y)^y)_(z^y))^(u_(y_v)) =y [9,1]
23 (((x ^y)_y)^(z_y))_(u^(v^((w^y)_(y^v6)))) = y[2,9,2,2,2]
24 (((x _y)^(z^y))_(u^(z^y)))^(v_y) =z^y [9,5]
25 (((x ^y)_(z_y))^(u_(z_y)))_(v^y) =z_y [5,9]
26 (((x ^y)_y)^(z_y))_(u^(v^(y^w))) = y [12,9,12,12,12]
27 ((x _(y_(x_z)))^(u_(y_(x_z))))_(x_z) =y_(x_z) [22,8]
28 (((x _y)^y)_(z^y))^(u_(v_(y_w))) = y [22,5,22,22,22]
29 (((x _(y_z))^(y_z))_(u^z))^(v_((y_z)_w)) =y_z [5,22]
30 (((x _(((y^z)_(u_z))^(v_(u_z))))^(((y^z)_(u_z))^(v_
(u_z))))_(w^(((y^z)_(u_z))^(v_(u_z)))))^(v6_(u_z)) =
((y^z)_(u_z))^(v_(u_z)) [14,22]
31 (((x _(((y^(z^(u^v)))_u)^(w_u)))^(((y^(z^(u^v)))_u)^(w
_u)))_(v6^(((y^(z^(u^v)))_u)^(w_u))))^(v7_u) =
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
152 Appendix A: Some Prover9 Proofs
((y^(z^(u^v)))_u)^(w_u) [21,22]
32 (((x ^(y_(z_(u_v))))_(y_(z_(u_v))))^(w_(y_(z_(u_v)))))
_u=y_(z_(u_v)) [22,9]
33 (((((x _y)^y)_(z^y))^(u^y))_(v^(u^y)))^y=u^y[1,17,1,1,1]
34 (((x ^(y^(z^u)))_z)^(v_z))_(w^(z^v6)) = z[12,20,12,12,12]
35 (((x _y)^y)_(z^(u^(y^v))))^(w_((v6_y)^(y_v7)))
=y [12,10,12,12,12,12]
36 (((((x ^y)_y)^(z_y))_(u_y))^(v_(u_y)))_y=u_y[2,27,2,2,2]
37 (((x ^(y^(z^u)))_z)^(v_z))_(w^((v6^z)_(z^v7)))
=z [12,11,12,12,12,12]
38 (((x ^(y^((z_u)^v)))_(z_u))^(w_(z_u)))_u=z_u [1,37]
39 (((((x _y)^y)_(z^y))^(u^y))_(v^(u^y)))^(w_y)
=u^y [1,18,1,1,1]
40 (((x _y)^y)_(z^y))^(((u_(y_v))^(y_v))_(w^(y_v))) = y
[15,1]
41 (((x ^y)_y)^(z_y))_(u^(v^(((w^y)_(y^v6))^v7))) = y[19,2]
42 (((x ^y)_(z_y))^(u_(z_y)))_(v^(w^(y^v6))) = z_y[19,25]
43 (((x ^y)_y)^(z_y))_(u^(v^y)) =y [1,41]
44 (((x ^y)_y)^(z_y))_(u^(v^(w^y))) = y [43,9,43,43,43]
45 (((x _y)^y)_(z^(u^(v^y))))^(w_(y_v6)) = y[43,29,43,43,43]
46 (((x ^(y^(z^u)))_z)^(v_z))_(w^(v6^z)) =z[43,38,43,43,43]
47 (((x ^y)_y)^(z_y))_(u^(v^((w^(v6^y))^v7))) = y [19,44]
48 (((x _y)^y)_(z^(u^((v^(w^y))^v6))))^(v7_(y_v8))
=y [19,45]
49 (((x _y)^(z^y))_(u^(z^y)))^(v_(w_(y_v6))) = z^y[32,24]
50 (((x ^(y_z))_(y_z))^(u_(y_z)))_y=y_z [40,2]
51 (((x _y)^y)_(z^y))^(u_((y_v)^(y_w))) = y[40,6,40,40,40,40]
52 (((x ^y)_(z_y))^(u_(z_y)))_z=z_y [40,11]
53 (((x ^y)_y)^(y_z))_(u^(y^v)) =y [40,16,51,51,51]
54 (((x ^y)_(z_y))^(u_(z_y)))_(v^z) =z_y [53,11]
55 (((x _y)^y)_(z^y))^(((u_y)^(y_v))_w) =y [52,1]
56 (((x ^(y^(z^u)))_z)^(v_z))_(((w^(v6^(z^v7)))_z)^(v8_z))
=z [34,52,34,34]
57 (((x ^y)_(z_y))^(u_(z_y)))_(v^(((w^y)_(z_y))^(v6_(z_y
)))) = z_y [14,54,14,14]
58 (((x ^(y^(z^u)))_z)^(v_z))_(w^(((v6^(v7^(z^v8)))_z)^
(v9_z))) = z [34,54,34,34]
59 (((x _y)^y)_(z^y))^y=y [2,55]
60 ((x _x)^(y_x))_(z^(x^u)) = x [59,12]
61 ((x _x)^(y_x))_(z^(u^(x^v))) = x [59,26]
62 ((x _x)^(y_x))_(z^(u^((v^x)_(x^w)))) = x [59,23]
63 (x _(y^x))^x=x [59,33,59,59,59]
64 (((x ^(y^(z^u)))_z)^(v_z))_((w^z)_(z^v6)) = z [59,37]
65 (((x ^(y^(z^u)))_z)^(v_z))_(w^z) =z [59,46]
66 ((x _x)^(y_x))_(z^(u^((v^(w^x))^v6))) = x [59,47]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix A: Some Prover9 Proofs 153
67 (((x ^y)_(z_y))^(u_(z_y)))_(v^(y^w)) =z_y [59,42]
68 (x _(y^x))^(z_x) =x [59,39,59,59,59]
69 (((x _y)_(x_y))^(z_(x_y)))_x=x_y [59,50]
70 (x _(y^(z^((u^x)_(x^v)))))^x=x [13,63]
71 (x _x)^x=x [59,63]
72 (x _(y^(z^((u^(v^x))^w))))^(v6_(x_v7)) = x [71,48]
73 ((x ^y)_(z_y))^(u_(z_y)) = (z _y)^(v_(z_y)) [71,30,57]
74 ((x ^(y^(z^u)))_z)^(v_z) =z^(w_z) [71,31,58]
75 (x _(y^(z^((u^x)_(x^v)))))^(w_x) =x [13,68]
76 ((x ^(y^z))_(u^(x^(y^z))))^z=x^(y^z) [43,68]
77 (x _x)^(y_x) =x [59,68]
78 (x _y)_x=x_y [69,77]
79 x_(y^(z^((u^(v^x))^w))) = x [66,77]
80 x_(y^(z^((u^x)_(x^v)))) = x [62,77]
81 x_(y^(z^(x^u))) = x [61,77]
82 x_(y^(x^z)) =x [60,77]
83 x^(y_(x_z)) =x [72,79]
84 x^(y_x) =x [75,80]
85 x^x=x [70,80]
86 ((B ^A)_(A^C))_A6=Aj((B_A)^(A_C))^A6=A [3,82,83]
87 ((x ^y)_(z_y))^(u_(z_y)) =z_y [73,84]
88 ((x ^(y^(z^u)))_z)^(v_z) =z [74,84]
89 (x _y)_(z^(y^u)) = x_y [67,87]
90 x_(y^x) =x [65,88]
91 x_((y^x)_(x^z)) =x [64,88]
92 x_x=x [56,88,88]
93 (x ^(y^z))^z=x^(y^z) [76,90]
94 (x ^y)_y=y [85,12,89]
95 x_(y_x) =y_x [85,36,94,84,78]
96 (x ^y)^(z_(u_(y_v))) = x^y [85,49,94]
97 (x _y)^y=y [92,28,96]
98 x^(y^x) =y^x [92,33,97,90,93]
99 x^((y_x)^(x_z)) =x [92,35,97,81]
100 ((x ^y)_(y^z))_y=y [91,95,91]
101 ((B _A)^(A_C))^A6=A [86,100]
102 ((x _y)^(y_z))^y=y [99,98,99]
103 2 [102,101]
Huntington Identity, Proof of Order Reversibility
Here it is proved that if a uniquely complemented lattice satis es ax-
iom (8) in Theorem 4.8.4, then complementation is antitone. In view of
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
154 Appendix A: Some Prover9 Proofs
Theorem 4.8.1(1), this implies Theorem 4.8.4(8).
Input le:
formulas(assumptions).
% Lattice Theory
x v y = y v x.
x ^ y = y ^ x.
(x v y) v z = x v (y v z).
(x ^ y) ^ z = x ^ (y ^ z).
x ^ (x v y) = x.
x v (x ^ y) = x.
% Complementation
x v x' = 1.
x ^ x' = 0.
% Unique Complementation
x v y != 1 | x ^ y != 0 | x' = y # label(Unique_complementation).
% Identity H82
(x ^ y) v (x ^ z) = x ^ ((y ^ (x v z)) v (z ^ (x v y))) # label(H82).
% Denial of order reversibility
A ^ B = A.
A' v B' != A' # answer(Order_reversibility).
end_of_list.
Prover9 proof in 3 seconds:
1 x_y=y_x [assumption]
2 x^y=y^x [assumption]
3 (x _y)_z=x_(y_z) [assumption]
4 (x ^y)^z=x^(y^z) [assumption]
5 x^(x_y) =x [assumption]
6 x_(x^y) =x [assumption]
7 x_x0= 1 [assumption]
8 x^x0= 0 [assumption]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix A: Some Prover9 Proofs 155
9 x_y6= 1jx^y6= 0jx0=y# Unique complementation [assumption]
10 (x ^y)_(x^z) =x^((y^(x_z))_(z^(x_y))) # H82 [assumption]
11 A^B=A [assumption]
12 A0_B06=A0# Order reversibility [assumption]
13 x^(y^z) =y^(x^z) [2,4,4]
14 x^(y_x) =x [1,5]
15 x_((x^y)_z) =x_z [6,3]
16 x_(x0_y) = 1_y [7,3]
17 x^1 =x [7,5]
18 x_0 =x [8,6]
19 0 _(x^y) =x^(y_(x0^(x_y))) [7,10,8,1,17]
20 A^(B^x) =A^x [11,4]
21 1 ^x=x [17,2]
22 0 _x=x [18,1]
23 x^(y_(x0^(x_y))) = x^y [19,22]
24 x^(y^x0) =y^0 [8,13]
25 1 _x= 1 [21,5]
26 x_(x0_y) = 1 [16,25]
27 0 ^x= 0 [22,5]
28 x_1 = 1 [14,21]
29 x^0 = 0 [27,2]
30 x^(y^x0) = 0 [24,29]
31 A^(A0_B0)6= 0 # Order reversibility [9,26,12]
32 x_(x^y)0= 1 [7,15,28]
33 x_(y^x)0= 1 [2,32]
34 B_A0= 1 [11,33]
35 B^(A0_B0) =B^A0[34,23,2,21]
36 A^(A0_B0) = 0 [35,20,30]
37 2# Order reversibility [36,31]
Huntington Identity, Proof of Distributivity
Here we prove Theorem 4.8.4(8).
Input le:
formulas(assumptions).
% Lattice Theory
x v y = y v x.
x ^ y = y ^ x.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
156 Appendix A: Some Prover9 Proofs
(x v y) v z = x v (y v z).
(x ^ y) ^ z = x ^ (y ^ z).
x ^ (x v y) = x.
x v (x ^ y) = x.
% Complementation
x v x' = 1.
x ^ x' = 0.
% Unique Complementation
x v y != 1 | x ^ y != 0 | x' = y # label(Unique_complementation).
% Identity H82
(x ^ y) v (x ^ z) = x ^ ((y ^ (x v z)) v (z ^ (x v y))) # label(H82).
% Denial of distributivity
(A ^ B) v (A ^ C) != A ^ (B v C) # answer(Distributivity).
end_of_list.
Prover9 proof in 85 seconds:
1 x_y=y_x [assumption]
2 x^y=y^x [assumption]
3 (x _y)_z=x_(y_z) [assumption]
4 (x ^y)^z=x^(y^z) [assumption]
5 x^(x_y) =x [assumption]
6 x_(x^y) =x [assumption]
7 x_x0= 1 [assumption]
8 x^x0= 0 [assumption]
9 x_y6= 1jx^y6= 0jx0=y# Unique complementation [assumption]
10 (x ^y)_(x^z) =x^((y^(x_z))_(z^(x_y))) # H82 [assumption]
11 (A ^B)_(A^C)6=A^(B_C) # Distributivity [assumption]
12 x_(y_z) =y_(x_z) [1,3,3]
13 x^(y^z) =y^(x^z) [2,4,4]
14 x^(y_x) =x [1,5]
15 x^((x_y)^z) =x^z [5,4]
16 x_(y^x) =x [2,6]
17 x_((x^y)_z) =x_z [6,3]
18 (x ^y)_(x^(y^z)) =x^y [4,6]
19 x^x=x [6,5]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix A: Some Prover9 Proofs 157
20 x_(x0_y) = 1_y [7,3]
21 x_(y_(x_y)0) = 1 [7,3]
22 x^1 =x [7,5]
23 x^(y^(x^y)0) = 0 [8,4]
24 x_0 =x [8,6]
25 x_y6= 1jy^x6= 0jy0=x [1,9]
26 (x ^y)_z6= 1jx^(y^z)6= 0j(x^y)0=z [4,9]
27 x^((y^(x_z))_(z^(x_y))) = (x ^z)_(x^y) [10,1]
28 0 _(x^y) =x^(y_(x0^(x_y))) [7,10,8,1,22]
29 x^(y^x) =y^x [19,4,2]
30 1 ^x=x [22,2]
31 0 _x=x [24,1]
32 x^(y_(x0^(x_y))) = x^y [28,31]
33 x^(y^(x_z)) =y^x [5,13]
34 1 _x= 1 [30,5]
35 x_(x0_y) = 1 [20,34]
36 0 ^x= 0 [31,5]
37 x^((y_x)^z) =x^z [14,4]
38 x^(y^(z_x)) = y^x [14,13]
39 x_1 = 1 [14,30]
40 x^0 = 0 [36,2]
41 x_((y^x)_z) =x_z [16,3]
42 x^(x_y)0= 0 [8,15,40]
43 x^(y_x)0= 0 [1,42]
44 x^(y_(z_x))0= 0 [3,43]
45 x^(y^(z_(x^y))0) = 0 [43,4]
46 x_(x^y)0= 1 [7,17,39]
47 x_(y^x)0= 1 [2,46]
48 x_(y_((x_y)^z)0) = 1 [46,3]
49 (x ^y)_(y^x) =x^y [29,18]
50 x^(y^(y^x)0) = 0 [2,23]
51 x_y6= 1jx^y6= 0jy0=x [2,25]
52 x00=x [7,25,2,8]
53 x^(y^x)06= 0jy^x=x [47,25,2,52]
54 x^(y^((x^y)0_z))6= 0j(x^y)0_z= (x^y)0[35,26]
55 x^((y^(z_x))_(z^(x_y))) = (x ^z)_(x^y) [1,27]
56 x^(y^(y^(x_z))0) = 0 [50,15,40]
57 x^(y^(z_(y^x))0) = 0 [49,44,4]
58 x^(x0_(y_(x_y)0)) =x^(y_(x_y)0) [21,32,2,30,1]
59 x^((y_x)^(x_z))0= 0 [56,37]
60 (x _y)^x06= 0jx_y=x [5,53,5]
61 (x _y)^y06= 0jy_x=y [1,60]
62 x0_(y^x)0= (y^x)0[32,54,23,47,2,30,1]
63 x0^(y^x)0=x0[62,5]
64 x0_(x_y)0=x0[5,62,1,5]
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
158 Appendix A: Some Prover9 Proofs
65 x_(y^(x_y0))0= 1 [62,48,2]
66 x0^(x_y)0= (x_y)0[5,63,2]
67 x0_(y_(x_z)0) =y_x0[64,12]
68 x^(y_(x_y)0) =x^(y_x0) [58,67]
69 (x _y)^(y_(x_y)0) =y [51,65,59,52]
70 x_(y0^(x_y))0= 1 [52,65]
71 x_((x_y)^y0)0= 1 [2,70]
72 x^(y_(x^y0))0= 0 [66,45]
73 x^(y_(y0^x))0= 0 [66,57]
74 x^(y0_(x^y))0= 0 [52,72]
75 x^(y0_(y^x))0= 0 [52,73]
76 x^((x^y)_y0)0= 0 [1,74]
77 x^((y^x)_y0)0= 0 [1,75]
78 x^(y_x0) =x^y [69,15,68]
79 x^(y_(z_x0)) =x^(y_z) [3,78]
80 (x _y)^y0=x^y0[71,78,2,30,2,15]
81 x^y06= 0jy_x=y [61,80]
82 (x ^y)_x0=y_x0[81,77,1,41]
83 (x ^y)_y0=x_y0[81,76,1,17]
84 (x ^(y_z))_y0=x_y0[33,82,83]
85 (x ^(y_z))_z0=x_z0[38,82,83]
86 (x ^y)_(x^z) =x^(y_z) [55,78,3,84,12,85,79]
87 2# Distributivity [86,11]
References
[1] W. McCune. Prover9. http://www.cs.unm.edu/~mccune/pro
ver9/, 2005{2007.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness
We have not found the items marked by *, which we quote from Bull. Sci.
Math. 15(1891), 53-68; Jahr. Fortschr. Math. 35(1904), 82; 53(1927), 44,
Huntington [1917], [1935] Yooneyama [1918] and Katri~ n ak [1959].
It is well known that with every relation of partial order, i.e., a binary
relation which satis es
(1) xx;
(2) xy&yx=)x=y;
(3) xy&yz=)xz;
is associated the relation <de ned by
(4) x<y ()xy&x6=y:
This relation satis es
(5) x6<x;
(6) x<y =)y6<x;
(7) x<y &y<z =)x<z ;
note that (6) implies (5).
Conversely, with any relation <ofstrict partial order, i.e., satisfying (5){
(7), is associated a partial order de ned by
(8) xy()x<y orx=y:
The correspondence (4), (8) is one-to-one. So
P1=f(1); (2);(3)g
159
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
160 Appendix B: Partially Ordered Sets and Betweenness
and
P2=f(6); (7)g
are independent sets of axioms for partially ordered sets orposets, i.e., sets
endowed with a partial order.
The founders of the theory of partially ordered sets are Peirce [1880,87],
Schr oder [1890-1905] and Hausdor [1914]; cf. Birkho [1948], Ch.I. foot-
note 1. As a matter of fact, in those early days the main interest was
concentrated on what we call today totally ordered sets orchains, that is,
sets endowed with a relation <which satis es (5){(7) and
(9) x6=y=)x<y ory<x:
Besides, in the years 1890-1930 the abstract relation <was not yet well
crystallized and many authors worked with <; > and = as three inter-
connected primitive concepts. So did Betazzi [1890]*, Burali-Forti [1893]*,
Stoltz and Gmeiner [1901]*, Shatunovski i [1904]*, Yooneyama [1918] and
Su [1927]*.
The rst axiom systems for totally ordered sets use the above axioms
and
(60) :(x<y &y<x);
(10) x<y =)x6=y;
(11) :(x6=y&x<y &y<x);
(12) x;y;z distinct &x<y &y<z =)x<z ;
(13) x<y =)z <y orx<z :
The very rst system seems to be
T1=f(60);(7);(9)g;
given by Vailatti [1892]*, while Huntington [1904-6], [1917] found the sys-
tems
T2=f(6); (7);(9)g;
T3=f(7); (9);(10)g;
T4=f(5); (9);(11); (12)g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 161
T5=f(60);(9);(12)g;
and proved that they are completely independent. Later on, Huntington
[1935] devised the system
T6=f(5); (9);(11); (13)g:
SoT1 T5are maximal independent subsystems of f(1);:::; (12)g, hence
they are logically equivalent to it, but it is not known whether there are also
other maximal independent subsystems. The complete existential theory
of system f(1);:::; (12)g was not studied.
Chains have been also characterized in terms of the relation of open
betweenness, de ned by
(14) (xyz)()x<y<z orz <y<x:
It is easy to check the following properties:
(15) x;y;z distinct & (xyz ) =)(zyx);
(16)x;y;z distinct = )(xyz) or (xzy ) or (yxz ) or (yzx) or (xzy ) or (zyx);
(17) x;y;z distinct = ) :((xyz ) & (xzy ));
(18) (xyz) =)x;y;z distinct;
(19) x;y;z;t distinct & (xyz ) & (yzt) = )(xyt);
(20) x;y;z;t distinct & (xyz ) & (ytz ) =)(xyt);
(21) x;y;z;t distinct & (xyz ) & (ytz ) =)(xtz);
(22) x;y;z;t distinct & (xyz ) & (xtz ) =)(xyt) or (xty );
(23) x;y;z;t distinct & (xyz ) & (xtz ) =)(xyt) or (tyz );
(24) x;y;z;t distinct & (xyz ) & (tyz ) =)(xtz) or (txz );
(25) x;y;z;t distinct & (xyz ) & (tyz ) =)(xty) or (txy );
(26) x;y;z;t distinct & (xyz ) & (tyz ) =)(xty) or (txz );
(27) x;y;z;t distinct & (xyz ) =)(xyt) or (tyz ):
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
162 Appendix B: Partially Ordered Sets and Betweenness
Huntington and Kline [1917] and Huntington [1924a] have proved that
the following axiom systems de ne totally ordered sets and are independent:
T7=f(15); (16); (17); (18); (19); (20)g;
T8=f(15); (16); (17); (18); (19); (23)g;
T9=f(15); (16); (17); (18); (19); (24)g;
T10=f(15); (16); (17); (18); (19); (25)g;
T11=f(15); (16); (17); (18); (19); (26)g;
T12=f(15); (16); (17); (18); (20); (22)g;
T13=f(15); (16); (17); (18); (20); (23)g;
T14=f(15); (16); (17); (18); (21); (23)g;
T15=f(15); (16); (17); (18); (21); (22); (24)g;
T16=f(15); (16); (17); (18); (21); (22); (25)g;
T17=f(15); (16); (17); (18); (21); (22); (26)g;
T18=f(15); (16); (17); (18); (27)g;
among which T18is completely independent. Van de Walle [1924] proved
that systems fT7;:::;T16gare also completely independent, while T17does
not share this property. Rosenbaum [1951] gave the following completely
independent axiom system for chains:
T19=f(15); (16); (18); (28); (29)g;
where the last two axioms are weaker variants of (19) and (20), respectively:
(28) (xyz) & (yzt) = )(xyt);
(29) (xyz) & (ytz ) =)(xyt):
The characterizations of chains by each system Ti;(i= 7;:::; 19) has
the following meaning. The total order <satis es Tiand conversely, if
a ternary relation on a set Tsatis es Ti, then for each pair of distinct
elementsa;b2T, the following relation <a;bis a total order on T:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 163
(30.1) a<a;bb&:(b< a;ba) &:(x< a;bx);
(30.2) x<a;ba() (xab);
(30.3) x<a;bb() (xab) or (abx) ;
(30.4) a<a;bx() (axb) or (abx) ;
(30.5) b<a;bx() (abx);
(30.6)x<a;by() (xab) & (xyb) or (xab) & (xby ) or
or (axb) & (axy ) or (abx) & (axy ):
Moreover, the betweenness relation (14) associated with <a;bcoincides with
the original ternary relation. On the other hand, if the starting point is
a chain (T;<) and a < b inT, then the construction (30) applied to the
betweenness relation (14) yields <a;b=<.
Another system, whose independence has not been studied, was given
by Moisil [1942], namely
T20=f(15); 19);(20); (31); (32)g;
where we have set
(31) :(xxy ) &:(xyx);
(32) x;y;z distinct = )(xyz) or (yzx) or (zxy ):
Cech [1936]* has characterized totally ordered sets with at least three
elements by the system
T21=f(32); (33); (34); (35); (36); (37); (38); (39)g;
where we have set
(33) (xyz) =)x6=z;
(34) (xyz) =)x6=y;
(35) (xyz) =)(zyx);
(36) (xyz) & (yzt) = )(xzt);
(37) (xyz) & (xzt) = )(xyt);
(38) (xyz) & (xtz ) &y6=t=)(xyt) or (xty );
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
164 Appendix B: Partially Ordered Sets and Betweenness
(39) (xyz) & (xyt) & z6=t=)(yzt) or (ytz ):
For every set of cardinality at least 3 and endowed with a ternary relation
which satis es system T21, there are exactly two relations of total order such
that the betweenness relation (14) associated with them coincides with the
original ternary relation (these two relations are dual to each other). This
property is \better" than the corresponding property of the total orders
<a;b.
Katri~ n ak [1959] has shown that system T21is not independent, while
systems
T22=f(32); (34); (35); (36); (37); (39)g;
T23=f(32); (33); (35); (36); (39)g;
T24=f(32); (34); (35); (37); (38); (39)g;
are independent and logically equivalent to T21.
Shepperd [1956] has characterized totally ordered sets in termes of the
relation of closed betweenness, de ned by
(40) xyz()xyzorzyx:
This is a specialization of Smiley and Transue's relation of lattice be-
tweennes, which can be used to de ne lattices; cf. Chapter 1, x4, (1.4.12).
The axiom system provided by Shepperd is
T25=f(41); (42); (43); (44); (45)g;
where we have set
(41) xyzoryzxorzxy;
(42) xyz&xzy=)y=z;
(43) xyz=)zyx;
(44) xyz&xzt=)yzt;
(45) xyz&yzt&y6=z=)xyt:
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 165
McPhee [1962] proved that systems
T26=f(46); (47); (48); (49)g;
T27=f(48); (49); (50)g;
T28=f(47); (51); (52)g;
are independent and logically equivalent to T25, where
(46) 9axya orxayoryaxoryxaoraxyorayx;
(47) yxz&ztx=)xty;
(48) yxz&tyx=)zxtorx=y;
(49) yxz&zxt&txy=)x=yorx=zorx=t;
(50) xyzoryzxoryxz;
(51) xyzorxzyoryzxoryxzorzxyorzyx;
(52) zxt&zyt&xzy&xty=)x=zory=zorx=tory=t:
A ternary relation which appears in geometry is the cyclic order ABC
between the points of an oriented circle. This relation means that the arc
ABC is positively oriented.
More generally, the relation of cyclic order is de ned on a totally ordered
set by
(53) xyz()x<y<z ory<z<x orz<x<y:
The following properties are easily checked:
(54) x;y;z distinct & xyz=) yzx;
(55)x;y;z distinct = ) xyzoryzxorzxyorzyx
oryxzorxzy;
(56) x;y;z distinct = ) :(xyz&xzy);
(57) xyz=)x;y;z distinct;
(58) x;y;z;t distinct & xyz&ytz=) xyt;
(59) x;y;z;t distinct & xyz&ytz=) xtz ;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
166 Appendix B: Partially Ordered Sets and Betweenness
(60) x;y;z;t distinct & xyz=) xytortyz:
Huntington [1924b] has proved that the following axiom systems de ne
totally ordered sets and are completely independent:
T29=f(54); (55); (56); (57); (58)g;
T30=f(54); (55); (56); (57); (59)g;
T31=f(54); (55); (56); (57); (60)g;
and if the support set Thas at least three elements, the system
T32=f(54); (56); (57); (60); (61)g;
where axiom (61) states that there exist distinct elements a;b;c satisfying
abc, de nes also chains.
The characterization of chains by each system Ti(i= 29;:::; 32) has
the following meaning. The total order <satis es Tiand conversely, if a
ternary relation on a set Tsatis es Ti, then for each element a2Tthe
following relation <ais a total order on Thavingaas least element:
(62.1) x<ay() axy;
(62.2) :(x< ax);
(62.3) x6=a=)a<ax&:(x< aa):
Moreover, the following two properties I and II hold:
I. The relation of cyclic order (53) associated with <acoincides with
the original ternary relation.
Proof. If the elements x;y;z 2Tsatisfy relation (53) corresponding to
<a, then they are distinct and by (62.3) they are also distinct from a. So
it follows by (54) and (58) that each of the double inequalities x <ay <a
z; y < az <axandz <ax<ayimplies xyz. Conversely, suppose xyz
holds. Then x;y;z are distinct by (57). If one of these elements is a, in
view of (54) we can suppose without loss of generality that x=a, hence
x=a <ay <azby (62), therefore the right-hand side of (53) holds, as
desired. Now suppose x;y;z are also distinct from a. Then (60) implies
xyaorayz; in the former case axy, again by (54). So x<ayory<az.
Since<ais a total order, there are three possibilities: 1) x <ay <az, or
2)x <ayandz <ay, or 3)y <axandy <az. The desired conclusion
is already established in case 1). Case 2) yields the subcases x <az <ay
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 167
andz <ax<ay; in each subcase the desired conclusion holds. Case 3) is
treated similarly.
II. If the starting point is a chain (T;<) having aas least element, then
the construction (62) applied to the original cyclic order (53) yields <a=<.
Proof. It follows by (62.1) and (53) that
y<ax() ayz()a<y<z ()y<z:
Another system, whose independence was not studied, was given by
Moisil [1942], namely
T33=f(54); (63); (64); (65)g;
where
(63) x;y;z distinct = ) xyzorxzy;
(64) : xxy;
(65) x;y;z distinct & xyz&xzt=)xyt:
Another relation which appears in geometry is the separation relation
ABCD between the points of a circle, de ned as follows: \the points AC
separate the points BD", or equivalently, \the sum of the unoriented arcs
ABC andADC is 2 ".
More generally, the separation relation is de ned on a totally ordered
set by
(66)xyzt()x<y<z<t ory<z<t<x orz <t<x<y or
ort<x<y<z ort<z<y<x orx<t<z <y or
ory<x<t<z orz <y<x<t:
The following properties are satis ed:
(67) xyzt =)x;y;z;t distinct;
(68) x;y;z;t distinct &xyzt =)yztx;
(69) x;y;z;t distinct = ) :(xyzt &xytz);
(70) x;y;z;t distinct &xyzt =) 9a;b;c;d distinct &abcd &dcba;
(71) x;y;z;t;u distinct &xyzt =)xuzt orxyzu;
(72) x;y;z;t distinct = )xyzt orxytz or:::ortzyx;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
168 Appendix B: Partially Ordered Sets and Betweenness
(73) x;y;z;t distinct &xyzt =)tzyx;
(74) x;y;z;t;u distinct &xyzt &xytu =)xyzu;
(75) x;y;z;t;u distinct &xyzt &xytu =)yztu;
(76) x;y;z;t;u distinct &xyzt &xytu =)xztu;
(77) x;y;z;t;u distinct &xyzt &xytu =)xyzu orxyuz;
(78) x;y;z;t;u distinct &xyzt &xyzu =)xztu orxzut;
(79) x;y;z;t;u distinct &xyzt &xyzu =)yztu oryzut;
(80)x;y;z;t;u distinct &xyzt &xyzu =)(xytu orxzut) & (xyut orxztu);
(81)x;y;z;t;u distinct &xyzt &xyzu =)(xytu oryztu) & (xyut oryztu);
(82)z;y;z;t;u distinct &xyzt &xyzu =)(xztu oryzut) & (xzut oryztu):
Huntington and Rosinger [1932]* have proved that the following axiom
systems de ne totally ordered sets with least element and are independent:
T34=f(67); (68); (69); (71); (72); (73)g;
T35=f(67); (68); (69); (72); (73); (75)g;
T36=f(67); (68); (69); (72); (73); (76)g;
T37=f(67); (68); (69); (72); (73); (74); (77)g;
T38=f(67); (68); (69); (72); (73); (74); (78)g;
T39=f(67); (68); (69); (72); (73); (74); (79)g;
T40=f(67); (68); (69); (72); (73); (74); (80)g;
T41=f(67); (68); (69); (72); (73); (74); (81)g;
T42=f(67); (68); (69); (72); (73); (74); (82)g;
and if the support Thas at least 4 elements, the system
T43=f(67); (68); (70); (83)g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 169
where axiom (83) states that there exist distinct elements a;b;c;d satisfying
abcd, de nes also chains.
The characterization of chains by each system Ti(i= 23;:::; 43) has
the following meaning. The total order <satis es Tiand conversely, if a
quaternary relation on a set Tsatis es Ti, then for each triple of distinct
elementsa;b;c2Tthe following relation <a;b;cis a total order on Thaving
aas least element:
(84.1) b<a;b;cc&:(c< a;b;cb&:(x< a;b;cx);
(84.2) x6=a=)a<a;b;cx&:(x< a;b;ca);
(84.3) x<a;b;cb()axbc;
(84.4) b<a;b;cx()abxc orabcx;
(84.5) x<a;b;cc()axbc orabxc;
(84.6) c<a;b;cx()abcx;
(84.7)x<a;b;cy() (axbc &axyb) or (axbc &axcy) or
or (abxc &abxy) or (abcx &abxy):
On the other hand, if the starting point is a chain (T;<) having a least
elementAand ifb;c2Tsatisfya < b < c, then the construction (84)
applied to the separation relation (66) yields <a;b;c=<.
Shepperd [1956] has characterized totally ordered sets in terms of the
relation of closed separation, de ned by
(85)xyzt()xyztoryztxorztxyor
ortxyzortzyxor
orxtzyoryxtzorzyxt:
His axiom system is
T44=f(86); (87); (88); (89); (90)g;
where
(86) xyzt orxzyt orxytz;
(87) xyzt orxytz()x=yorz=t;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
170 Appendix B: Partially Ordered Sets and Betweenness
(88) xyzt =)txyz;
(89) xyzt =)yztx;
(90) xyzt &xuyz &y6=z=)xuyt:
See systems T45-T50afterP13.
Problem 1 in Birkho [1948] asks for a characterization of partially
ordered sets in terms of the closed betweenness relation
(40) xyz()xyzorzyz:
The rst answer to this problem was given by Altwegg [1950], who devised
the system
P3=f(43); (91); (92); (93); (94); (95); (96)g;
where we have set
(91) xxx;
(92) xyz=)xxy ;
(93) xyx=)x=y;
(94) xyz&yzu&y6=z=)xyu;
(95) x;y;z;t distinct &xyz&ytt=)(xyt &:tyz) or (:xyt &tyz);
(96)8n(xi 1xi 1xi&:xi 1xixi+1) (i= 1;:::; 2n&
&x2n+1 =x0) =)x2nx0x1:
The characterization of posets by system P3has the following meaning.
Firstly, the closed betweenness relation of every poset satis es P3. Con-
versely, suppose Pis a set endowed with a ternary relation which satis es
P3. Two elements x;y2Pare said to be comparable ifxyyorxxy, and
connected provided they are linked by a connected sequence, that is, a se-
quencex=x0;x1;:::;x n=ywith the property that any two consecutive
terms are comparable. In particular any two comparable elements are con-
nected. The comparability relation is clearly symmetric and also re
exive
by (91), hence so is connectedness. Besides, the concatenation of two con-
nected sequences is also connected, again by (91). Therefore connectedness
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 171
is an equivalence relation, whose cosets will be called the connected com-
ponentsP(2I) of P.yNow from each connected component Pwhich is
not a singleton we choose two distinct comparable elements a;b, Further-
more suppose x;y2Pare distinct comparable elements. Then x;y2P
for some (unique) 2I. Sinceb;x2P, there is a connected sequence
b;x1;:::;x n;xand this implies that the sequence a;b;x1;:::;x n;x;y is
also connected, hence so is the sequence
(97) a;b;b;x1;x1;:::;x n;xn;x;x;y:
By replacing all possible occurrences in (97) of three consecutive terms
z;z;z byz;z, sequence (97) is reduced to a connected subsequence (possibly
the same) of the form
(98)a=zr0;b;:::;z r1;:::;zri 1;:::;z ri;zri;:::;z ri+1;:::
:::;zrs:::;x;z rs+1=y;
where each subsequence zri 1;:::;z riconsists of distinct terms. One proves
that the parity of sdoes not depend on the choice of the starting sequence
x1;:::;x)n. This enables one to set
(99) xa;bj2Iy()s0 (mod 2):
One proves that this relation is a partial order on P. Moreover, the
closed betweenness (40) associated with abj2Icoincides with the origi-
nal ternary relation. On the other hand, if the starting point is a connected
poset (P ), which means there is only one connected component, and the
construction (99) is applied to the closed betweenness relation (40), then
abj2Pis either the original relation or its dual.
Sholander [1952], using the paper by Altwegg, obtained a shorter system
equivalent to P3, namely
P4=f(100); (101); (102)g;
where
(100) xyx()x=y;
(101) xyz&ytu=)zytoruyx;
(102)8n3nodd &x1;:::;x ndistinct &x1x1x2&:::&xn 1xn 1xn
=)xn 1xnx1orxnx1x2or9i1in 2 &xix1+1×1+2:
yThis concept of connectedness is borrowed from graph theory.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
172 Appendix B: Partially Ordered Sets and Betweenness
Morinaga and Nishig ori [1953], unaware of the papers by Altwegg and
Sholander, characterized posets by systems
P5=f(91); (95); (103); (104); (105); (107); (108); (109)g;
P6=f(91); (103); (104); (105); (107); (108); (110); (111)g;
P7=f(91); (103); (105); (106); (107); (112); (113)g;
where
(103) x6=z&xyz=)zyx;
(104) x6=y&xyz=)xxy;
(105) x6=z&xyz&xzy=)y=z ;
(106) xyzoryzxorzxy=)xyy;
(107) y6=z=) :(xyz &xzy);
(108) x6=z&xyz&yzt&y6=z=)xzt;
(109)8nx1;:::;x 2n+1 distinct &x1x2x2&x2x3x3&:::
&x2n+1x1x1=) 9i1i2n 1 &xixi+1xi+2;
(110)8nx1;:::;x 2n+1 distinct &x1x2y1&x2x3y2&:::
&x2n+1x1y2n+1 =) 9i1i2n 1 &xixi+1xi+2;
(111)x;y;z;t distinct &xyz&ytu=)(xyt &:tyz) or (:xyt &tyz);
(112)x;y;z;t distinct &xyz& (ytu ortuyoruyt)
=)(xyt &:tyz) or (:xyt &tyz);
(113)8nx1;:::;x 2n+1 distinct & (x iyixi+1or &yixi+1xiorxi+1xiyi)
(i= 1;:::; 2n+ 1 &x2n+2 =x1)
=) 9i1i2n 1 &xixi+1xi+2:
The construction and the proofs for P5 P7are more complicated than
forP4, but the idea is the same.
The systems P3 P7are independent.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 173
Morinaga and Nishig ori [1953] have also characterized partially ordered
sets with least element 0 by the independent axiom systems
P8=f(91); (95); (103); (104); (106); (108); (114); (115); (116)g;
P9=f(91); (103); (104); (106); (108); (111); (117)g;
P10=f(91); (103); (104); (106); (108); (117); (118)g;
where
(114) 0xx;
(115) x0y=)x= 0 ory= 0;
(116)x1;x2;x3distinct &x1x2y1&x2x3y2&x3x1y3
=)x1x2x2orx2x3x1orx3x1x2;
(117) xyz=)0xyor 0yz;
(118) x;y;z;t distinct &xyz&xzt=)yzt:
In the same paper the open betweenness relation (14)is used to charac-
terize posets by the system
P11=f(15); (17); (119); (120)g;
where
(119)x;y;z;t;u distinct & (xyz ) & ((ytu) or (tuy ) or (uyt))
=)((xyt) & :(tyz )) or (:(xyt) & (tyz ));
(120)8nx1;:::;x 2n+1 distinct &y1;:::;y 2n+1 distinct &x2n+3 =x2&
((xixi+1yi) or (x i+1yixi) or (y ixixi+1)) (i= 1;:::; 2n+ 1 &x2n+2 =x1)
=)xi;xi+1;yidistinct (i = 1;:::; 2n+ 1) &
9j1j2n+ 1 & (x jxj+1xj+2);
posets with least element 0 by the system
P12=f(15); (17); (120); (121)g;
where
(121) x;y;z distinct & ((xyz ) or (yzx) or (zxy )) =)(0xy) or (0yx);
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
174 Appendix B: Partially Ordered Sets and Betweenness
and posets with 0 and 1 by the system
P13=f(15); (17); (121); (122)g;
where
(122) x;y;z distinct & ((xyz ) or (yzx) or (zxy )) =)(0x1):
The following independent axioms systems for totally ordered sets in
terms of closed betweenness:
T45=f(43); (44); (93); (95)g;
T46=f(100); (101); (102); (123)g;
T47=f(41); (100); (101)g;
T48=f(91); (103); (106); (108); (108); (118); (124); (125)g;
T49=f(91); (95); (103); (106); (124); (125)g;
were given by Altwegg [1950] ( T46;T47), Sholander [1952] ( T46;T47),
and Morinaga and Nishig ori [1953] ( T48;T49;) where we have set
(123) 8x8y8z9txty orytzorztx;
(124) x6=y=)xyy;
(125) x;y;z distinct = )xyzoryzxorzxy:
In a totally ordered set T, the closed betweenness relation has the prop-
erty that for any x;y;z 2T, exactly one of these elements, say (x;y;z ),
is between the two others. Then the following properties hold:
(126) (x;x;y ) =x;
(127) ( (x;y;z ); (x;y;t);u) = ( (z;t;u);x;y );
(128) x6= (x;y;z )6=z=) (x;t;y ) =yor (y;t;z ) =y:
Sholander [1952] characterized totally ordered sets in terms of the ternary
operation by the system
T50=f(126); (127); (128)g;
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 175
to the e ect that chains satisy T50and conversely, if a set Tendowed
with a ternary operation satis es T50, then the ternary relation de ned
byxyz() (x;y;z ) =yful ls system T47, henceTbecomes a totally
ordered set.
The relation of closed betweenness has been also used by Padmanabhan
[1966] to characterize totally ordered sets among lattices, as we describe
below.
Consider the following predicates in a lattice L:
A(a;b;x) ()axborbxa;
B(a;b;x) () (a^x)_(b^x) =x= (a_x)^(b_x);
C(a;b;x) () (a^x)_(b^x) =x= (a^b)_x;
C0(a;b;x) () (a_x)^(b_x) =x= (a_b)^x;
D(a;b;x) ()a^bxa_x:
A(a;b;x) is closed betweenness, denoted by axbin this Appendix. B(a;b;x)
was introduced by Glivenko as the lattice-theoretic characterization of met-
ric betweenness and adapted by Pitcher and Smiley as lattice betweenness
in arbitrary lattices; cf. Theorems 1.4.2 and 2.2.2. The relations CandC0
are characterizations of metric betweenness due to Blumenthal and Ellis.
Remark The following implications are obvious: A=)B=)C=)D
andB=)C0=)D.
Theorem I The following conditions are equivalent in a lattice L:
(i)Lis a chain;
(ii)D=)A:
Proof: (i)=)(ii): Suppose the elements a;b;x of a chainLsatisfya^b
xa_b. Thenaxborbxaaccording as aborba.
(ii)=)(i): Takea;b2L. SinceD(a;b;a) is true, it follows that
A(a;b;a) also holds, which reduces to aborba. 2
Theorem II A lattice L is a chain if and only if conditions A,B,C,C0,D
are equivalent in L.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
176 Appendix B: Partially Ordered Sets and Betweenness
Proof: Immediate from Theorem I and the Remark. 2
Besides, the same paper provides similar characterizations of modular
and distributive lattices within the class of all lattices.
The notation xyzused in this Appendix for closed betweenness coincides
with the notation axbin Chapters 1 and 2 for lattice betweenness. In view
of Theorem II, this is not so dramatic !
References
Altwegg, M.
1950. Zur Axiomatik der teilweise gordneten Mengen. Comment.
Math. Helv. 24, 149-155.
Betazzi, R.
1890.* Teoria delle Grandezze. E. Spoeri, Pisa.
Birkhoff, G.
1948. Lattice Theory. Amer. Math. Soc. Coll. Publ., New York.
Burali-Forti, C.
1893.* Sulla teoria delle grandezze. Riv. Mat. 3, 76-101.
Cech, E.
1936. Budove Mno ziny. Praha.
Clay, R.E.
1969. Sole axioms for partially ordered sets. Logique Anal. (N.S.) 12,
361-375.
Harary, F.
1961. A very independent axiom system. Amer. Math. Monthly 68,
159-162.
Hausdorff, F.
1914. Grundz uge der Mengenlehre. Leipzig.
Huntington, E.V.
1904-6. The continuum as a type of order: an exposition of the modern
theory. Ann. Math. 6, 151-184; 7, 15-43.
1917. Complete existential theory of the postulates for serial order.
Bull. Amer. Math. Soc. 23, 276-289.
1924a. Sets of complete independent postulates for cyclic order. Proc.
Nat. Acad, Sci. 10,74-78.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 177
1924b. A new set of postulates for betweenness, with a proof of com-
plete independence. Trans. Amer. Math. Soc. 26, 257-282.
1935. Inter-relations among the principal types of order. Trans. Amer.
Math. Soc. 38, 1-9.
Huntington, E.V.; Kline, J.R.
1917. Sets of independent postulates for betweenness. Trans. Amer.
Math. Soc. 18, 301-325.
Huntington, E.V.; Rosinger, K.E.
1932. Postulates for separation of point pairs (reversibility order on a
closed line). Proc. Amer. Acad. Arts Sci. (Boston) 67, 61-145.
Katri ~nak, T.
1959. Pozn amka k usporiadnym mno zina. Acta Fac. Rer. Natur.
Univ. Comenianae (Math.) 4, 291-299.
Kempe, A.B.
1890. On the relation between the logical theory of classes and the
geometrical theory of points. Proc. London Math. Soc. 21, 147-182.
Lihov a, J.
2000. Strict order betweenness. Acta Univ. M. Belii Ser. Math. No.8,
27-33. MR 2002f:06002.
Mac Neille, H.M.
1937. Partially ordered sets. Trans. Amer. Math. Soc. 42, 416-460.
McPhe, J.A.
1962. Axioms for betweenness. J. Math. Soc. 37, 112-116.
Moisil, Gr.C.
1942. Curs de analiz a general a. Univ. Bucure sti.
Morinaga, K.; Nishig ori, N.
1952-3. On the axioms of betweenness. J. Sci. Hiroshima Univ. (A)
16, 177-222; 399-408.
Padmanabhan, R.
1966. On some ternary relations in lattices. Colloq. Math. 15, 195-198.
Peirce, C.S.
1880,87. On the algebra of logic. Amer. J. Math. 3, 15-27; 7, 180-202.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
178 Appendix B: Partially Ordered Sets and Betweenness
Rosenbaum, I.
1951. A new system of completely independent postulates for between-
ness. Bull. Amer. Math. Soc. 57, 279.
Russell, B,
1903.* Principles of Mathematics. Cambridge.
Sakai, Sh.
1965.* Systems of postulates for a ternary relation characterizing a
partial order. Rep. Fac. Sci. Shizuoka Univ. 1, no.1, 1-4. MR 33#7276.
Schroder, E.
1890-1905. Vorlesungen uber die Algebra der Logik. Leipzig; vol.I,
1890; vol.II, 1891, 1905; vol.III, 1895. Reprint: Chelsea Publ. Co. Bronx,
New York, 1966.
Shatunovski i, S.O.
1904.* On the postulates de ning the concept of magnitude (Russian).
Mem. Math. Section Russian Soc. Naturalists (Odessa), 26, 21-25.
Shepperd, J.A.H.
1956. Transitivities of betweenness and separation and the de nitions
of betweenness and separation groups. J. London Math. Soc. 31, 240-248.
Sholander, M.
1952. Trees, lattices, order and betweenness. Proc. Amer. Math. Soc.
3, 369-381.
Stoltz, O.; Gmeiner, J.A.
1901.* Theoretische Arithmetik. Leipzig.
Su, B.
1927.* Geometrical proof of the independence of a system of postulates
concerning equaliy and inequality. T^ ohoku Math. J. 28, 282-286.
Trombetta, M.
1983. A system of axioms for the relation of betweenness (Italian).
Rend. Inst. Mat. Univ. Trieste 15, no.1-2, 96-107. MR 86k:06004.
Vailatti, G.
1892.* Sui principi fundamentali della geometria della retta. Riv. Mat.
2, 71-75.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix B: Partially Ordered Sets and Betweenness 179
Van de Walle, W.E.
1924. On the complete independence of the postulates for betweenness.
Trans. Amer. Math. Soc. 26, 249-256.
Yooneyama, K.
1918. Sets of independent postulates concerning equality and inequality
(theory of three unde ned relations). T^ ohoku Math. J. 14, 171-283.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
This page intentionally left blankThis page intentionally left blank
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix C: Quasilattices
Quasilattices were introduced by Padmanabhan [1971]. An algebra
(A;_;^) of type (2,2) is called a quasilattice if both _and^are semi-
lattice operations such that the natural partial order relation determined
by_enjoys the substitution property with respect to ^and vice-versa; that
is, i
Q1x_x=x;
Q2x_y=y_x;
Q3x_(y_z) = (x _y)_z ;
Q4x_y=y=)(x^z)_(y^z) =y^z;
and their duals hold. The class of quailattices is equational, because con-
dition Q4 is equivalent to
Q5 ((x _y)^z)_(x^z) = (x _y)^z
and dually. For taking y:=x_yin Q4 we obtain Q5, while if x_y=ythen
Q5 reduces to (y ^z)_(x^z) =y^z. The system of axioms Q consisting
of Q1{Q3, Q5 and their duals is independent; most of this fact was proved
by Padmanabhan (op. cit.), while the independence of Q3 (and of its dual)
was established by Chandran [1979]. The paper by Padmanabhan provides
also examples of quasilattices that are not lattices.
Recall (Chapter 1, x3) that an identity f=gis called regular if the
sets of variables occurring in the two sides of the equation are the same; for
instance, any identity valid in a semilattice is regular (cf. Lemma 1.3.2).
The identities in system Q are regular and valid in any lattice. Even more,
Padmanabhan proved that the set Q implies all the regular identities true
in any lattice. In other words, quasilattices capture the essence of regular
identities valid in all lattices. The proof is via a representation theorem for
quasilattices as the sum of a direct system of lattices. The essential tool
is the concept of a partition function in algebras, which is due to Plonka
[1968].
181
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
182 Appendix C: Quasilattices
The following two theorems are worth mentioning.
The class of quasilattices is the smallest equational class of algebras of
type (2,2) including both the class of lattices and the class of semilattices
(viewed as algebras (A; _;^) with _=^).
The class of quasilattices is not one-based, but can be de ned by two
identities.
Exercises
1. Derive
L23x^(x_y) =x_(x^y)
from Q.
2. A plethora of new axiom systems for lattices are obtained by the following
procedure. Let f=gbe any non-regular identity true in all lattices. Then
Q[ ff =ggis an equational basis for lattice theory.
3. The equivalence between the two distributivity laws D^
1and D_
1is valid
in the class of quasilattices as well.
Chandran, V.R.
1979. A note on Padmanabhan's paper \Regular identities in lattices".
Pure Appl. Math. Sci. 10, no.1-2, 13-15.
Padmanabhan, R.
1971. Regular identities in lattices. Trans. Amer. Math. Soc. 158,
179-188.
Padmanabhan, R.; Penner, P.
1999. Structures of free n quasilattices. Algebra Colloq. 6, 249-260.
Plonka, J.
1968. Some remarks on sums of direct systems of algebras. Fund. Math.
62, 301-308.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix D: Lukasiewicz-Moisil Algebras
In 1940 Moisil created the algebraic counterparts of the Lukasiewicz 3-
valued and 4-valued logics, under the name of Lukasiewicz algebras. Then
Moisil introduced the n-valued Lukasiewicz algebras and later on, the -
valued Lukasiewicz algebras, where is the order type of an arbitrary linear
order. These algebras have been much studied, both for their own sake and
for their interest in logic as well as in switching circuit theory, as shown
by Moisil and his school. That is why today we refer to these algebras as
Lukasiewicz-Moisil algebras. They include as a particular case the well-
known Post algebras, which in their turn generalize Boolean algebras.
A concept also due to Moisil is that of a De Morgan algebra. This term
designates a bounded distributive lattice endowed with a unary operation
Nwhich satis es N(x_y) =Nx^NyandNNx =x, henceN(x^y) =
Nx_Ny. Ann-valued Lukasiewicz-Moisil algebra is a De Morgan algebra
equipped with an increasing sequence '1'2 'n 1of lattice
endomorphisms which satisfy
'i(x)^N'i(x) = 0 (i = 1;:::;n 1);
'i(Nx) =N'n i(x) (i = 1;:::;n 1);
'i'j(x) ='j(x) (i;j = 1;:::;n 1);
'i(x) ='i(y) (i= 1;:::;n 1) =)x=y:
The bibliography compiled below selects papers devoted to the axiomat-
ics of Lukasiewicz-Moisil algebras; most of them refer to 3{valued LM al-
gebras. In particular Petcu [1968] obtained a 4{basis and a 3{basis, while
Becchio [1978] de ned 3{valued LM algebras in terms of an implication
operation. Much more about Lukasiewicz-Moisil algebras can be found in
the monograph by Boicescu et al [1991].
183
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
184 Appendix D: Lukasiewicz-Moisil Algebras
Abad, M.; Figallo, A.
1984. Characterization of three-valued Lukasiewicz algebras. Rep.
Math. Logic 18, 47-59.
Becchio, D.
1973. Sur les d e nitions des alg ebres trivalentes de Lukasiewicz donn ees
par A. Monteiro. Logique et Analyse 63-64; 339-344.
1978. Logique trivalente de Lukasiewicz. Ann. Sci. Univ. Clermont-
Ferrand 16, 38-89.
Boicescu, V.; Filipoiu, A.; Georgescu, G.; Rudeanu, S.
1991. Lukasiewicz-Moisil Algebras. North-Holland, Amsterdam.
Cignoli, R.
1969. Algebras de Moisil de ordin n. PhD Thesis, Univ. Nac. del Sur,
Bah a Blanca.
Cignoli, R.; Monteiro, A.
1965. Boolean elements in Lukasiewicz algebras. II. Proc. Japan Acad.
41, 676-680.
Moisil, Gr.C.
1940. Recherches sur les logiques non-chrysippiennes. Ann. Sci. Univ.
Jassy 26, 431-466 = [1972], 195-232.
1941a. Notes sur les logiques non-chrysippiennes. Ann. Sci. Univ.
Jassy 27, 86-98 = [1972], 233-243.
1941b. Contributions a l' etude des logiques non-chrysippiennes.
I. Un nouveau syst eme d'axiomes pour les alg ebres lukasiewicziennes
t etravalentes. C.R. Acad. Sci. Roumanie 5, 289-293 = [1972], 283-286.
1960. Sur les id eaux des alg ebres lukasiewicziennes trivalentes. An.
Univ. C.I. Parhon Bucure sti, Ser. Acta Logica 3, 83-95 = [1972], 244-258.
1972. Essais sur les Logiques Non-Chrysippiennes. Ed. Acad.
R.S.Roumanie, Bucarest.
1972a. Les axiomes des alg ebres de Lukasiewicz n-valentes. [1972], 288-
310.
1972b. Sur les alg ebres de Lukasiewicz n-valentes. [1972], 311-324.
Monteiro, A.
1963. Sur la d e nition des alg ebres de Lukasiewicz trivalentes. Bull.
Math. Soc. Sci. Math. Phys. R.P.Roumaine (NS) 7(55), 3-12 = Notas de
L ogica Matem atica No.21, Inst. Mat., Univ. Nac. del Sur, Bah a Blanca.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix D: Lukasiewicz-Moisil Algebras 185
Monteiro, A.; Monteiro, L.
1996. Axiomes ind ependants pour les alg ebres de Nelson, de
Lukasiewicz trivalentes, de De Morgan et de Kleene. In: A. Monteiro,
Unpublished Papers. I. Notas de L ogica Matem atica No.40, Inst. Mat.,
Univ. Nac. del Sur (INMABB-CONICET), Bah a Blanca.
Monteiro, L.
1963. Axiomes ind ependants pour les alg ebres de Lukasiewicz triva-
lentes libres. Bull. Math. Soc. Sci. Math. Phys. R.P.Roumaine (NS)
7(55), 199-202 (1964) = Notas de L ogica Matem atica, Inst. Mat., Univ.
Nac. del Sur, Bah a Blanca.
1969. Sur le principe de d etermination de Moisil dans les alg ebres de
Lukasiewicz trivalentes. Bull. Math. Soc. Sci. Math. R.P.Roumaine
13(61), 447-448.
Petcu, A.
1968. The de nition of the trivalent Lukasiewicz algebras by three
equations. Rev. Roumaine Math. Pures Appl. 13, 247-250.
Rudeanu, S.
1994. On the axiomatics of Lukasiewicz-Moisil algebras. An. S ti. Univ.
Ovidius (Constant a) 2, 152-159.
Sicoe, C.
1967a. Note asupra algebrelor Lukasiewicz multivalente. Stud. Cerc.
Mat. 19, 1203-1207.
1967b. On many-valued Lukasiewicz algebras. Proc. Japan Acad. 43,
725-728.
1967c. A characterization of Lukasiewicz algebras. I. II. Proc. Japan
Acad. 43, 729-732; 733-736.
1968. Sur la d e nition des alg ebres lukasiewicziennes polyvalentes. Rev.
Roumaine Math. Pures Appl. 13, 1027-1030.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
This page intentionally left blankThis page intentionally left blank
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix E: Testing Associativity
It turns out that in the proof of the independence of an axiom system,
checking associativity is usually the most dicult point. We list below sev-
eral papers which provide suggestions in order to facilitate this task.
Boccioni, D.
1960. Condizioni di distributivit a ed associativit a unilaterali. Rend.
Sem. Mat. Univ. Padova 30, 178-193. MR 22A#4652.
1963a. Condizioni di mutua distributivit a con ripetizioni. Rend. Sem.
Mat. Univ. Padova 33, 60-84. MR 28#50.
1963b. Condizioni independenti ed equivalenti a quella di mutua dis-
tributivit a. Rend. Sem. Mat. Univ. Padova 33, 91-98. MR28#51.
Ferrero, G.; Ferrero Cotti, C.
1975. Come veri care la propriet a associativit a. Boll. Un. Mat. Ital.
(4), 11, 322-329. MR55#3113.
Frazer, W.D.
1973. On testing a binary operation for associativity. Combinatorial
algorithms (Courant Comput. Sci. Sympos., No.9, 1972), 77-90. Algorith-
mic Press, New York. MR50#13353.
Szasz, G.
1953. Die Unabh angigkeit der Assoziativit atsbedingungen. Acta Sci.
Math. Szeged 15, 20-28. MR15,No.1,p.95.
1954. Uber die Unabh angigkeit kommutativer multiplikativer Struk-
turen. Acta Sci. Math. Szeged 15, 130-142. MR15, p.773.
187
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
This page intentionally left blankThis page intentionally left blank
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix F: Complete Existential Theory and
Related Concepts
The bibliography below, which has no claim to be exhaustive, collects pa-
pers devoted to complete existential theories and complete independence
of certain sets of postulates. Most of the papers refer to partially ordered
sets, lattices or Boolean algebras. Other elds are ring theory (Gilmer Jr
[1966]), binary relations (Petre [2002]), binary operations (Robinson [1971])
and associative semirings (Rudeanu and Vaida [2004]).
The papers by Avann [1964] and Gilmer Jr [1966] are only close to
complete existential theories. The notice by Ingraham [1923] comments
complete independence in general.
Last but not least, let us recall that the father of complete existential
theory is Moore [1910].
Avann, S.P.
1964. Dependence of niteness conditions in distributive lattices.
Math. Z. 85, 245-256.
Diamond, A.H.
1933. The complete existential theory of the Whitehead-Russell set of
postulates for the algebra of logic. Trans. Amer. Math. Soc. 35, 940-948;
correct. ibid. 36(1934), 893.
Dines, L.L.
1914-5. Complete existential theory of She er's postulates for Boolean
algebras. Bull. Amer. Math. Soc. 21(1914-5), 183-188.
Dubreil-Jacotin, M.L.; Lesieur, L.; Croisot, R.
1953. Le cons sur la Th eorie des treillis, des Structures Alg ebriques
Ordonn es et des Treillis G eom etriques. Gauthier-Villars, Paris.
Gilmer Jr, R.W.
Eleven nonequivalent conditions on a commutative ring. Nagoya Math.
J. 26, 183-194.
189
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
190 Appendix F: Complete Existential Theory and Related Concepts
Huntington, E.V.
1917. Complete existential theory for serial order. Bull. Amer. Math.
Soc. 23, 276-280.
1924a. Sets of completely independent postulates for cyclic order. Proc.
Nat. Acad. Sci. USA 10, 74-78.
1924b. A new set of postulates for betweenness, with a proof of com-
plete independence. Trans. Amer. Math. Soc. 26, 247-282.
Ingraham, M.H.
1923. Certain limitations of the value of the complete independence of
a set of postulates. Bull. Amer. Math. Soc. 29, 199-200.
MacNeille, H.M.
1937. Partially ordered sets. Trans. Amer. Math. Soc. 42, 416-460.
Moore, E.H.
1910. Introduction to a Form of General Analysis (New Haven Colloq.,
1906). New Haven.
Padmanabhan, R.
1969. Implications among some link axioms in lattice theory. J. Madu-
rai Univ. 1, No.1, 27-40.
Petre, G.
2002. Teoria complet existent ial a a propriet at ilor relat iilor binare. Gaz.
Mat. 20(99), 226-238.
Robinson, D.F.
1971. A catalogue of binary relations. New Zealand Math. Mag. 8,
2-11.
Rosenbaum, I.
1951. A new system of completely independent postulates for between-
ness. Bull. Amer. Math. Soc. 57, 279.
Rudeanu, S.
1964. Logical dependence of certain chain conditions in lattice theory.
Acta Sci. Math. (Szeged) 25, 209-218.
1973. Elemente de Teoria Mult imilor. Univ. Bucure sti.
Rudeanu, S.; Vaida, D.
2004. Semirings in operations research and computer science: more
algebra. Fund. Inform. 61, 61-85.
Taylor, J.S.
1917. Complete existential theory of Bernstein's set of postulates for
Boolean algebras. Ann. Math. 19, 64-69.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Appendix F: Complete Existential Theory and Related Concepts 191
1920. She er's set of ve independent postulates for Boolean algebras
in terms of the operation \rejection" made completely independent. Bull.
Amer. Math. Soc. 26, 449-454.
Van de Walle, W.E.
1924. On the complete independence of the postulates for betweenness.
Trans. Amer. Math. Soc. 26, 249-356.
Two related concepts were introduced by Church [1925]. Let A=
fA1;:::;A ngbe a system of axioms. An axiom of A, sayA1, can be weak-
ened with respect to A, if there is an axiom B1weaker than A1such that
the system fB1;A2;:::;A ngis equivalent to A. SettingA2&:::&An=B,
this means
(1) A1=)B1; B16=)A1; B1&B=)A1:
It is easy to see that A1cannot be weakened with respect to Ai
:A1=)B. For if the latter implication were consistent with (1), then
fromB1&B=)A1and:B=)A1we would infer B1=)A1. The
opposite implication is obtained by taking B1:=A1_ :B .
The system Ais said to be irredundant if it is independent and no axiom
ofAcan be weakened with respect to A. Therefore an independent set of
postulates is irredundant i :Ai=)Ajfor every two distinct axioms of
A.
The latter remark shows in particular that irredundancy is inconsistent
with complete independence.
For other remarks see Gehman [1926].
Church, A
1925. On irredundant sets of postulates. Trans. Amer. Math. Soc. 27,
318-328.
Gehman, H.M.
1926. On irredundant sets of postulates. Bull. Amer. Math. Soc. 32,
159-161.
As is well known, in a model which proves the independence of an
axiomAof a system A, the failure of Ameans that Adoesn't hold for
certain values of its variables. According to Harary [1961], axiom Aisvery
independent if its independence can be proved by a model in which Anever
holds, that is, it fails for all possible values given to its variables. The
system Aitself is said to be very independent if all of its axioms are so, and
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
192 Appendix F: Complete Existential Theory and Related Concepts
absolutely independent if for every subset SofAthere is a model in which
A Sholds and each axiom of Snever holds.
It is proved that the following system of axioms for equivalence relations
is absolutely independent: re
exivity, symmetry and transitivity restricted
to pairwise distinct elements x;y;z .
Harary, F.
1961. A very independent axiom system. Amer. Math. Monthly 68,
159-162.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography
Abbott, J.C.
1967a. Implicational algebras. Bull.Math. Soc. Sci. Math. Roumanie
11(59), 3-23.
1967b. Semi-Boolean algebra. Mat. Vesnik 4(19), 177-198.
Arai, Y.; Is eki, K.
1965. Axiom systems of B-algebra. II. Proc. Japan Acad. 41, 908-910.
Bennett, A.A.
1930. Semi-serial order. Amer. Math. Monthly 39, 289-295.
1933. Solution of Huntington's unsolved problem in Boolean algebra.
Bull. Amer. Math. Soc. 39, 289-195.
Beran, L.
1976. Three identities for ortholattices. Notre Dame J. Formal Logic
17, 251-252.
1982. Boolean and orthomodular lattices – a short characterization via
commutativity. Acta Univ. Carolinae Math. Phys. 23, 25-27.
Bernstein, B.A.
1914. A complete set of postulates for the logic of classes, expressed in
terms of the opration \exception" and a proof of the independence of a set
of postulates due to Del Re. Univ. Calif. Publ. Math. 1, 87-96.
1915-1916. A simpli cation of the Whitehead-Huntinton set of postu-
lates for Boolean algebras. Bull. Amer. Math. Soc. 22, 458-459.
1916. A set of four independent postulates for Boolean algebras. Trans.
Amer. Math. Soc. 17, 50-52.
1924. Complete sets of representatives of two-element algebras. Bull.
Amer. Math. Soc. 30, 24-30.
1931. Whitehead and Russell's theory of deduction as a mathematical
science. Bull. Amer. Math. Soc. 37, 480-488.
193
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
194 Bibliography
1933. Simpli cation of the set of four postulates for Boolean algebras
in terms of rejection. Bull. Amer. Math. Soc. 39, 783-787.
1934. A set of four postulates for Boolean algebra, in terms of the im-
plicative operation. Trans. Amer. Math. Soc. 36, 876-884.
1936. Postulates for Boolean algebra involving the operation of com-
plete disjunction. Ann. Math. 37, 317-325.
1939. Sets of postulates for Boolean groups. Ann. Math. 40, 420-422.
1944. Postulate-sets for Boolean rings. Trans. Amer. Math. Soc. 55,
393-400.
1950. A dual symmetric de nition of Boolean algebra, free from postu-
lated special elements. Scripta Math. 16, 157-161.
Birkhof, G.
1948. Lattice Theory. Amer. Math. Soc. Coll. Publ., New York. First
edition, 1940. Second edition, 1948; third printing, 1961. Third edition,
1967.
Birkhoff, G.; Kiss, S.A.
1947. A ternary operation in distributive lattices. Bull. Amer. Math.
Soc. 53, 749-752.
Birkhoff, G.; von Neumann, J.
1936. The logic of quantum mechanics. Ann. Math. 37, 823=843 = J.
von Neumann, Collected Papers. Pergamon Press, 1961, vol.IV, 105-125.
Birkhoff, G.D.; Birkhoff, G.
1946. Distributive postulates for systems like Boolean algebras. Trans.
Amer. Math. Soc. 60, 3-11.
Blumenthal, L.M.; Bumcrot, R.J.
1962. Betweenness relations in lattices. Notices Amer. Math. Soc. 9,
111.
Boole, G.
1847. The Mathematical Analysis of Logic. Cambridge.
1854. An Investigation into the Laws of Thought. London. = Reprint,
Open Const. Publ. Co, Chicago 1940.
Bosbach, B.
1969. Komplement are Halbgruppen. Axiomatik und Arithmetik. Fund.
Math. 64, 257-287.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 195
1977.` group cone and Boolean algebra. A common one-identity ax-
iom. Contributions to Universal Algebra, Colloq. Math. Soc. Janos Bolyai,
vol.17. North-Holland, Amsterdam, 41-56.
Burris, S.; Sankappanavar, H.P.
1981. A Course in Universal Algebra. Springer-Verlag, New York.
Byrne, L.
1946. Two brief formulations of Boolean algebra. Bull. Amer. Math.
Soc. 52, 269-272.
1948. Boolean algebra in terms of inclusion. Amer. J. Math. 70, 139-
143.
1951. Short formulations of Boolean algebra, using ring operations.
Canad. J. Math. 3, 31-33.
Carloman, A.G.
1976. Axiomatica de algebra de Boole. Gac. Mat. (Madrid) (1) 28,
No.1-2, 19-24.
Chen, C.C.; Gr atzer, G.
1969. On the construction of complemented lattices. J. Algebra 11,
56-63.
Cheremisin, A.I.
1958. Solution of a problem of Birkho . (Rusian). Uch. Zap. Ivanovsk.
Gos. Ped. In-ta 18, 205-207.
Church, A.
1952. Review of C. Pereira's paper \Sobre el algebra de la l ogica de
Schr oder". J. Symb. Logic 17, 154.
Couturat, L.
1911. L'Alg ebre de la Logique. II- eme ed. Scientia, Paris.
Croisot, R.
1951. Axiomatique des lattices distributives. Canad. J. Math. 3, 24-27.
Dahn, B.I.
1998. Robbins algebras are Boolean: A revision of McCune's computer-
generated solution of Robbins problem. J. Algebra 208, 526-532.
Day, A.
1977. Splitting lattices generate all lattices. Algebra Universalis 7,
163-169.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
196 Bibliography
Dedekind, R.
1987. Uber Zerlegung von Zahlen durch ihre gr ossten gemeinsamen
Teiler. Festchrift Tech. Hochschule Braunschweig, 1-40. = Gesammelte
Math. Werke, vol.II, 103-147.
Del Re, A.
1907. Lezioni sulla Algebra della Logica. Napoli.
1911. Sulla independenza dei postulati dell'algebra della logica. Rend.
Acad. Napoli (3) 17, 450-458.
Diamond, A.H.
1934a. The complete existential theory of the Whitehead-Huntington
set of postulates for the algebra of logic. Trans. Amer. Math. Soc. 35,
940-948; corrections ibid. 36(1935), 893.
1934b. Simpli cation of the Whitehead-Huntington set of postulates for
the algebra of logic. Bull. Amer. Math. Soc. 40, 599-601.
Diamond, A.H.; McKinsey, J.C.C.
1947. Algebras and their subalgebras. Bull. Amer. Math. Soc. 53,
959-962.
Dicker, R.M.
1963. A set of independent axioms for Boolean algebra. Proc. London
Math. Soc. 13, 20-30.
Diego, A.; Suarez, A.
1966. Two sets of axioms for Boolean algebras. Portugal. Math. 23,
139-145.
Dines, L.L.
1914-15. Complete existential theory of She er's postulates for Boolean
algebras. Bull. Amer. Math. Soc. 21, 183-188.
Drasickova, H.
1966. […] Mat.-Fyz. Casopis 16, 1-13.
Dubreil-Jacotin, M.-L.; Lesieur, L.; Croisot, R.
1953. Le cons sur le Th eorie des Treillis, des Structurea Alg ebriques
Ordonn ees et des Treilis G eom etriques. Gauthier-Villars, Paris.
Ellis, D.
1949. Notes on the foundations of lattice theory. Publ. Math. Debrecen
1, 205-208.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 197
Felscher, W.
1957. Ein unsymmetrisches Assoziativ-Gesetz in der Verbandstheorie.
Arch. Math. 8, 171-174.
1958. Beziehungen zwischen Verbands- ahnliche Algebren und geregel-
ten Mengen. Math. Ann. 135, 369-387.
Felscher, W.; Klein-Barmen, F.
1959 Zur Axiomatik der kommmutativen Halbverb anden. Arch. Math.
10, 7.
Ferentinou-Nicolacopoulou, J.
1968. Sur une axiomatique des treillis distributifs. Praktika Akad.
Athen. 43, 64-69 (1969).
Freese, R.; Jezek, J.; Nation, J.B.
1995. Free Lattices. Math. Surveys Monographs #42. Amer. Math.
Soc., Providence.
Frink, O.
1926. The operations of Boolean algebras. Ann. Math. 27, 477-490.
1941. Representations of Boolean algebras. Bull. Amer. Math. Soc.
47, 775-776.
Gerrish, F.
1978. The independence of Huntington's axioms for Boolean algebra.
Math. Gaz. 62, no.419, 35-40.
Glivenko, V.
1936. G eom etrie des syst emes de choses norm ees. Amer. J. Math. 58,
799-828.
Goetz, A.
1971. On various Boolean structures in a given Boolean algebra. Publ.
Math. Debrecen 18, 103-107.
Goodstein, R.L.
1963. Boolean Algebra. Pergamon Press, Oxford; MacMillan Co., New
York.
Gould, M.I.; Gr atzer, G.
1967. Boolean extensions and normal subdirect powers of nite univer-
sal algebras. Math. Z. 99, 16-25.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
198 Bibliography
Gratzer, G.
1962. On Boolean functions. (Notes on lattice theory. II). Rev. Math.
Pures Appl. 7, 693-697.
1971. Lattice Theory: First Concepts and Distributive Lattices. Free-
man & Co., San Francisco.
1978. General Lattice Theory. Academic Press, New York; 2nd edition,
Birkh auser Verlag, 1998.
1979. Universal Algebra. Second edition, Springer-Verlag, New York-
Heidelberg-Berlin. First edition, Van Nostrand Co., Princeton, 1968.
2007. Two problems that shaped a century of lattice theory. Notices
Amer. Math. Soc. 54, 696-707.
Grau, A.A.
1947. Ternary Boolean algebra. Bull. Amer. Math. Soc. 53, 567-572.
Guting, R.
1971. Ein neues Axiomensystem f ur Boolesche Verb ande. J. Reine
Angew. Math. 251, 212-220.
Hahn, O.
1909. Zur Axiomatik des logischen Gebietkalkul. Arch. Sys. Phil. 15,
345-347.
Hammer (Iv anescu), P.L.
1959. She erian algebras. Bull. Math. Soc. Sci. Math. Phys.
R.P.Roumaine 3(51), 289-303.
Hashimoto, J.
1951. A ternary operation in lattices. Math. Japon. 2, 49-52.
Hedlikov a, J.; Katri ~nak, T.
On a characterization of lattices by the betweenness relation – a problem
of M. Kolibiar. Algebra Universalis 28, 389-400.
Henle, P.
1932. The independence of the postulates for logic. Bull. Amer. Math.
Soc. 38, 409-414.
Higman, G.; Neumann, B.H.
1952. Groups as groupoids with one law. Publ. Math. Debrecen 2,
215-221.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 199
Hoberman, S.: McKinsey, J.C.C.
1937. A set of postulates for Boolean algebras. Bull. Amer. Math. Soc.
43, 588-592.
Huntington, E.V.
1904. Sets of postulates for the algebra of logic. Trans. Amer. Math.
Soc. 5, 288-309.
1932. A new set of independent postulates for the algebra of logic, with
special reference to Whitehead and Russell's Principia Mathematica. Proc.
Nat. Acad. Sci. USA 18, 179-180.
1933a. New sets of independent postulates for the algebra of logic,
with special reference to Whitehead and Russell's Principia Mathematica.
Trans. Amer. Math. Soc. 35, 274-304. Corrections ibid. 557-558, 971.
1933b. A simpli cation of Lewis and Langford's postulates for Boolean
algebra. Mind 42, 203-207.
Imai, Y.; Is eki, K.
1967. On axioms of B-algebra. Portugal. Math. 26, 23-30.
Iseki, K.
1965a. Algebraic formulations of propositional calculi. Proc. Japan
Acad. 41, 803-807.
1965b. Axiom systems of B-algebra. Proc. Japan Acad. 41, 808-811.
1965c. A characterization of Boolean algebra. Proc. Japan Acad. 41,
893-897.
1968. A simple characterization of Boolean rings. Proc. Japan Acad.
44, 923-924.
1972. On axioms of Boolean algebra. Proc. Japan Acad. 48, 101-102.
Iseki, K.; ^Ohashi, S.
1970. Axiom systems of distributive lattices. Proc. Japan Acad. 46,
409-410.
Isobe, K.
1973. A note on the de nition of a Boolean ring (Japanese). Mem.
Kitami Inst. Tech. 4, 321-325.
Jipsen, P.; Rose, H.
1992. Varieties of Lattices. Lecture Notes #1553, Springer-Verlag.
Kalman, J.A.
1959. On the postulates for lattices. Math. Ann. 137, 362-370.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
200 Bibliography
1960. Equational completeness and families of sets closed under sub-
traction. Indag. Math. 22, 402-405.
1968. A two-axiom de nition for lattices. Rev. Roumaine Math. Pures
Appl. 13, 669-670.
Kalmbach, G.
1983. Orthomodular Lattices. Academic Press, London/New York.
Katri ~nak, T.
1959. Pozn amka k usporiadanym mno zina. Acta Fac. Rer. Nat. Univ.
Comenian. 4, 291-299.
1961. O jednej ot azke charakteriz ace sv azu pomocou tern arnej op eracie.
Acta Fac. Rer. Nat. Univ. Comenian. 6, 335-341.
Kelly, D.; Padmanabhan, R.
1989. Variety-independence in lattice theory. Algebra Universalis 26,
380-394.
2002. Self-dual lattice identities. Algebra Universalis 48, 413-426.
2003. Self-dual bases for varieties of orthomodular lattices. Algebra
Universalis 50, 141-147.
2004. Irredundant self-dual bases for self-dual lattice varieties. Algebra
Universalis 52, 501-517.
2005. Orthomodular lattices and permutable congruences. Algebra Uni-
versalis 53, 227-228.
2007. Another independent self-dual basis for the trivial variety. Alge-
bra Universalis 57, 497-499.
Kelly, L.M.
1952. The geometry of normed lattices. Duke Math. J. 19, 661-669.
Kempe, A.B.
1890. On the relation between the logical theory of classes and the ge-
ometrical theory of points. Proc. London Math. Soc. 21, 147-182.
Kimura, N.
1950. Independency of the axioms of lattices. Rep. K odai Math. Sem.
2, 14.
Klein-Barmen, F.
1932. Uber einen Zerlegungssatz in der Theorie abstrakten
Verkn upfungen. Math. Ann. 106, 114-130.
1934. Beitr age zur Theorie der Verb ande. Math. Z. 39, 227-239.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 201
1939. Axiomatische Untersuchungen zur Theorie der Halbverb ande und
Verb ande. Deutsche Math. 4, 32-43.
1940. Uber eine weitere Verallgemeinerung des Verbandsbegri . Math.
Z. 46, 472-480.
1959. Uber modi zierte Gruppen- und Verbandsaxiome. Arch. Math.
10, 101-103.
Kobayasi, M.
1943. On the axioms of the theory of lattices. Proc. Imp. Acad. Tokyo
19, 6-9.
Kolibiar, M.
1956a. Charakteriz acia sv azu pomocou tern arnej oper acie. Mat.-Fys.
Casopis Sloven. Akad. Vied 6, 10-13.
1956b. On the axiomatics of modular lattices (Russian). Czechoslovak
Math. J. 6(81), 381-386.
1958. Charakterisierung der Verb ande durch die Relation "zwischen".
Z. Math. Logik u. Grundl. Math. 4, 89-100.
Kolibiar, M.; Marcisov a, T.
1974. On a question of J. Hashimoto. Mat. Casopis Sloven. Akad.
Vied 24, 179-185.
Kurosh, A.G.
1935. Durchschnittdarstellungen mit irreduziblen Komponente in Ringe
und sogenante Dualgruppen. Mat. Sb. 42, 613-616.
Lalan, V.
1950. Equations fonctionnelles dans un anneau bool eien. C.R. Acad.
Sci. Paris 230, 603-605.
Lewis, C.I.; Langford, C.H.
1932. Symbolic Logic. New York, London. Second ed., Dover Publ.,
New York, 1954.
Lisovik, L.P.
1997. On the axiomatization of Boolean algebras (Russian). Ukr. Mat.
Zh. 49, 937-942.
Lowig, H.F.J.
1969. Review to ^Ohashi [1968]. Math. Rev. 39, #1369.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
202 Bibliography
MacNeille, H.M.
1937. Partially ordered sets. Trans. Amer. Math. Soc. 42, 416-460.
Malliah, C.
1968. Independent postulates of Boolean algebras. Pi Mu Epsilon 4,
378-379.
1971. Short de nitions of lattices. Rev. Roumaine Math. Pures Appl.
16, 333-339.
1978. Sets of axioms for distributive lattices. Rev. Roumaine Math.
Pures Appl. 23, 187-195.
Martin, L.H.
1965. Two ternary algebras and their associated lattices. Amer. Math.
Monthly 72, 1088-1091.
Matusima, Y.
1952. On some problems of Birkho . Proc. Japan Acad. 28, 19-24.
McCune, W.
1997. Solutions of the Robbins problem. J. Automated Reason. 19,
263-276.
McCune, W.; Padmanabhan, R.
1996a. Automated Deduction in Equational Logic and Cubic Curves.
Lecture Notes Arti cial Intelligence 1095, Springer-Verlag, Berlin.
1996b. Single identities for lattice theory and weakly associative alge-
bras. Algebra Universalis 36, 436-449.
McCune, W.; Padmanabhan, R.; Veroff, R.
2003. Yet another single law for lattices. Algebra Universalis 50, 165-
169.
McCune, W.; Veroff, R.; Fittelson, B.; Harris, K.; Feist, A.
2002. Short single axiom for Boolean algebra. J. Automat. Reason. 29,
1-16.
McKenzie, R.N.
1970. Equational bases for lattice theories. Math. Scand. 27, 24-38.
Mendelsohn, N.S.; Padmanabhan, R.
1972. A single identity for Boolean groups and Boolean rings. J. Alge-
bra 20, 78-82.
1975. Minimal identities for Boolean groups. J. Algebra 34, 451-457.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 203
Meredith, C.A.
1969. Equational postulates for the She er stroke. Notre Dame J. For-
mal Logic 10, 266-270.
Meredith, C.A.; Prior, A.N.
1968. Equational logic. Notre Dame J. Formal Logic 9, 212-226.
Miller, D.G.
1952. Postulates for Boolean algebras. Amer. Math. Monthly 59, 93-
96.
Moisil, Gr.C.
1959. Asupra simpli c arii circuitelor cu tranzistori, a celor cu tuburi
electronice si a celor cu criotroane. Studii Cerc. Mat. 10, 7-67.
Montague, R.; Tarski, J.
1954. On Bernstein's self dual set of postulates for the algebra of logic.
Proc. Amer. Math. Soc. 5, 310-311.
Moore, E.H.
1910. Introduction to a Form of General Analysis. New Haven Coll.,
1906. Yale Univ. Press, New Haven.
Morgado, J.
1970. Some characterizations of Boolean rings. An. Acad. Brasil. Ci.
42, 641-644.
Newman, M.H.A.
1941. A characterization of Boolean lattices and rings. J. London Math.
Soc. 16, 256-272.
Nieminen, U.J.
1976. A ternary operation characterization of semi-Boolean algebras.
C.R. Acad. Bulgare Sci. 29, 303-305.
Nishig ori, N.
1954. A note on lattice segments. J. Sci. Hiroshima Univ. (A) 18,
123-127.
^Ohashi, S.
1968. On de nitions of Boolean rings and distributive lattices. Proc.
Japan Acad. 44, 1015-1017.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
204 Bibliography
Ore, O.
1935. On the foundations of abstract algebra. I. Ann. Math. 36,
406-437.
Padmanabhan, R.
1966. On axioms for semi-lattices. Canad. Math. Bull. 9, 357-358.
1966b. On some ternary relations in lattices. Coll. Math. 15, 195-198.
1968. A note on Kalman's paper. Rev. Roumaine Math. Pures Appl.
13, 1149-1152.
1969a. Inverse loops as groupoids with one law. J. London Math. Soc.
(2)1, 203-206.
1969b. Two identities for lattices. Proc. Amer. Math. Soc. 20, 409-
412.
1971. Regular identities in lattices. Trans. Amer. Math. Soc. 158,
179-188.
1972. On identities de ning lattices. Algebra Universalis 1, 359-361.
1977. Equational theory of algebras with a majority polynomial. Alge-
bra Universalis 7, 273-275.
1981. A rst order proof of a theorem of Frink. Algebra Universalis 13,
397-400.
1983. A self dual equational basis for Boolean algebras. Canad. Math.
Bull. 26, 9-12.
Padmanabhan, R.; McCune, W.
1995. Single identities for ternary Boolean algebras. Comput. Math.
Appl. 29, No.2, 13-16.
Padmanabhan, R.; McCune, W.; Veroff, R.
2007. Lattice laws forcing distributivity under unique complementation.
Houston J. Math. 33, 391-401.
Padmanabhan, R.; Penner, P.
2004. Semilattice operations generated by lattice terms. Order 21, 257-
263.
Padmanabhan, R.; Quackenbush, R.W.
1973. Equational theories of algebras with distributive congruences.
Proc. Amer. Math. Soc. 41, 373-377.
Peirce, C.S.
1880-84. On the algebra of logic. Amer. J. Math. 3(1880), 15-57;
7(1884), 180-202.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 205
Pereira, R.C.
1951. Sobre el algebra de la l ogica de Schr oder. Rev. Mat. Hisp. Amer.
(Ser. 4) 11, 222-239.
Petcu, A.
1964. Short de nitions of lattices, using the associative and absorption
laws. Stud. Cerc. Mat. 16, 1265-1280 (Romanian) = Rev. Roumaine
Math. Pures Appl. 10(1965), 339-355.
1967. De nirea algebrelor booleene si a semilaticilor cu ajutorul a dou a
axiome-identit at i. Stud. Cerc. Mat. 19, 891-895.
1971. O caracterizare global a a laticilor si semilaticilor. Bul. Inst.
Petrol, Gaze si Geol. 27, 265-268.
Phillips, J.D.; Vojt echovsk y, P.
2005. Linear groupoids and the associated wreath products. J. Sym-
bolic Comput. 40, 1106-1125.
Pic, Gh.
1969. Contributions a l'axiomatique des treillis distributifs. Studia
Univ. Babe s-Bolyai 14, No.1, 3-9.
Picu, C.I.
1982. Caracterizarea laticilor distributive prin echivalent e sau
implicat ii. Stud. Cerc. Mat. 34, 367-369.
Pixley, A.F.
1963. Distributivity and permutablility of congruence relations in equa-
tional classes of algebras. Proc. Amer. Math. Soc. 14, 105-109.
1971. The ternary discriminator function in universal algebra. Math.
Ann. 191, 167-180.
Ploscica, M.
1996. On a characterization of distributive lattices by betweenness re-
lation. Algebra Universalis 35, 249-255.
Ponticopoulos, L.
1962. Une contribution a la th eorie axiomatique des treillis. Bull. Soc.
Math. Gr ece 3, 12-18.
Potts, D.H.
1965. Axioms for semi-lattices. Canad. Math. Bull. 8, 519.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
206 Bibliography
Riecan, J.
1957. K axiomatike modul arnych sv azov. Acta Fac. Rer. Nat. Univ.
Comenian. 2, 257-262.
Rudeanu, S.
1959. Independent systems of axioms in lattice theory. Bull. Math.
Soc. Sci. Math. Phys. R.P.Roumaine 3(51), 475-488.
1961. On the de nition of Bolean algebras by means of binary opera-
tions (Russian). Rev. Math. Pures Appl. 6, 171-183.
1962. Asupra unor sisteme de postulate ale algebrelor booleene. Com.
Acad. R.P.Rom^ ane 12, 893-899.
1963. Axiomele Laticilor si ale Algebrelor Booleene. Ed. Acad.
R.P.Rom^ ane, Bucure sti.
1964. Logical dependence of certain chain conditions in lattice theory.
Acta Sci. Math. (Szeged) 25, 209-218.
1974. Boolean Functions and Equations. North-Holland, Amsterdam;
American Elsevier, New York.
Rudeanu, S.; Vaida, D.
2004. Semirings in operations research and computer science: more al-
gebra. Fund. Inform. 61, 61-85.
Ruedin, J.
1966. Sur les groupo des distributifs ayant un el ement neutre d'un c^ ot e
et sur l'axiomatique des treillis. C.R. Acad. Sci. Paris 263, 559-562.
1966-67. Groupo des distributifs et axiomatique des treillis. S eminaire
Dubreil-Pisot. Alg ebre et Th eorie des Nombres. 20, No.4.
1967. Distributivit e et axiomatique des treillis. C.R. Acad. Sci. Paris
265, 812-815.
1967-68. Axiomatique des treillis. S eminaire Dubreil-Pisot. Alg ebre et
Th eorie des Nombres. 21, No.9.
1968. Groupo des pseudo-distributifs a gauche et axiomatique des treil-
lis. C.R. Acad. Sci. Paris 266, 399-401.
Salii, V.N.
1988. Lattices with Unique Complements. Transl. Math. Monographs,
Amer. Math. Soc., Providence.
Sampathkumar, E.
1963. A set of minimum number of postulates for a Boolean algebra.
Proc. Indian Acad. Sci. 57, 254-258.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 207
Schroder, E.
1890-1905. Vorlesungen uber die Algebra der Logik. Leipzig. Vol.I,
1890; vol.II, 1891, 1905; vol.III, 1895. Reprint Chelsea Publ. Co. Bronx,
New York, 1966.
Sheffer, H.M.
1913. A set of ve independent postulates for Boolean algebras, with
applications to logical constants. Trans. Amer. Math. Soc. 14, 481-488.
Sholander, M.
1951. Postulates for distributive lattices. Canad. J. Math. 3, 28-30.
1952. Trees, lattices, order and betweenness. Proc. Amer. Math. Soc.
3, 369-381.
1953. Postulates for Boolean algebras. Canad. J. Math. 5, 460-464.
Sicoe, C.
1966. Axiom systems of B-algebra. Proc. Japan Acad. 42, 867-870.
Sioson, F.M.
1964. Equational bases of Boolean algebras. J. Symbolic Logic 29, 115-
124.
1965a. On Newman algebras. I. II. Proc. Japan Acad. 41, 31-34; 35-39.
1965b. A remark on Newman algebra. Kyungpook Math. J. 6, 59-67.
1967. Natural equational bases for Newman and Boolean algebras.
Compositio Math. 17, 299-310.
Smiley, M.F.; Transue, W.R.
1943. Applications of transitivites of betweenness in lattice theory. Bull.
Amer. Math. Soc. 49, 280-287.
Soboci nski, B.
1962. Six new sets of independent axioms for distributive lattices with
0 and 1. Notre Dame J. Formal Logic 3, 187-192.
1972a. Certain sets of postulates for distributive lattices with the con-
stant elements. Notre Dame J. Formal Logic 13, 119-123.
1972b. An abbreviation of Croisot's axiom-system for distributive lat-
tices with 1. Notre Dame J. Formal Logic 13, 139-141.
1972c. A new formalization of Newman algebra. Notre Dame J. Formal
Logic 13, 255-264.
1972d. An equational axiomatization of associative Newman algebras.
Notre Dame J. Formal Logic 13, 265-269.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
208 Bibliography
1972e. A semi-lattice theoretical characterization of associative New-
man algebras. Notre Dame J. Formal Logic 13, 283-285.
1973. A note on Newman's algebra system. Notre Dame J. Formal
Logic 14, 129-133.
1975a. A new postulate-system for modular lattices. Notre Dame J.
Formal Logic 16, 81-85.
1975b. A short postulate-system for ortholattices. Notre Dame J. For-
mal Logic 16, 141-144.
1976a. A short equational axiomatization of modular ortholattices.
Notre Dame J. Formal Logic 17, 311-316.
1976b. A short equational characterization of orthomodular lattices.
Notre Dame J. Formal Logic 17, 317-320.
1979. Equational two axiom bases for Boolean algebras and some lattice
theories. Notre Dame J. Formal Logic 20, 865-875.
Sorkin, Yu.I.
1951. Independent systems of axioms de ning lattices (Russian). Ukr.
Mat. Zh. 3, 85-97.
1954. On the embedding of latticoids and lattices (Russian). Dokl.
Akad. Nauk 95, 931-934.
1962. Constructing lattices by three identity-axioms (Russian). Uchen.
Zap. Mosk. Gos. Ped. In-ta No.8 (1962), 61-66.
Stabler, R.E.
1941. Sets of postulates for Boolean rings. Amer. Math. Monthly 48,
28-29.
Stamm, E.
1911. Beitrag zur Algebra der Logik. Montash. Math. 22, 137-149.
Stone, M.H.
1935a. Subsumption of Boolean algebras under the theory of rings.
Proc. Nat. Acad. Sci. 21, 103-105.
1935b. Postulates for Boolean algebras and generalized Boolean alge-
bras. Amer. J. Math. 57, 703-732.
1936. The theory of representations for Boolean algebras and general-
ized Boolean algebras. Trans. Amer. Math. Soc. 40, 37-111.
Sudkamp, T.A.
1976. A proof of Soboci nski's conjecture concerning a certain set of
lattice-theoretical formulas. Notre Dame J. Formal Logic 17, 615-616.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Bibliography 209
Szasz, G.
1952. On the independence of a postulate system for the distributive
lattices. Math. Ann. 124, 291-293.
1963. On independent systems of axioms for lattices. Publ. Math. (De-
brecen) 10, 108-115.
Tamura, S.
1970. Axioms for Boolean rings. Proc. Japan Acad. 46, 121-123.
1975. Two identities for lattices, distributive lattices and modular lat-
tices with a constant. Notre Dame J. Formal Logic 16, 137-140.
Tarski, A.
1938. Einige Bemerkungen zur Axiomatik der Boole'schen Algebra.
C.R. S eances Soc. Sci. Lettres Varsovie (Class III) 31, 33-35.
1968. Equational logic and equational theories of algebras. Contribu-
tions to Mathematical Logic. (Colloq. Hannover 1966), North-Holland,
275-288.
Taylor, J.S.
1917. Complete existential theory of Bernstein's rst set of four postu-
lates for Boolean algebras. Ann. Math. 19, 64-69.
1920a. A set of ve postulates for Boolean algebras in terms of the
operation \exception". Univ. Calif. Publ. Math. 1, 241-248.
1920b. She er's set of ve independent postulates for Boolean algebra
in terms of the operation \rejection" made completely independent. Bull.
Amer. Math. Soc. 26, 449-454.
Trevisan, G.
1951. Su una questione relativa alle strutture distributive. Rend. Ac-
cad. Sci. Fis. Mat. Napoli (IV)18, 144-145.
Vaida, D.
1957. Caracterizarea structurilor modulare prin sisteme cu axiome in-
dependente. Stud. Cerc. Mat. 8, 457-466.
Van Albada, P.J.
1964. A self-dual system of axioms for Boolean algebra. Proc. Nederl.
Akad. Wet. Ser. A 67, 377-381 = Indag. Math. 26, 377-381.
Vassiliou, Ph.
1950. A set of postulates for distributive lattices. Publ. Nat. Tech.
Univ. Athens no.5, 3.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
210 Bibliography
Veroff, R.
2000. Short 2-bases for Boolean algebra in terms of the She er stroke.
Tech. Rep. TR-CS-2000-25, Comput. Sci. Dept., Univ. New Mexico.
Albuquerque, NM.
2001. Lattice Theory. http://www.cs.unm.edu/~vero /LT/, 2001.
Wang, Sh.Ch.
1953. An axiom system for propositional calculus. J. Chinese Math.
Soc. 2, 267-274.
Whitehead, A.N.
1898. A Treatise on Universal Algebra, with Applications.Cambridge
Univ. Press.
Whiteman, A.
1937. Postulates for Boolean algebra in terms of ternary rejection. Bull.
Amer. Math. Soc. 43, 293-298.
Wiener, N.
1917. Certain formal invariances in Boolean algebra. Trans. Amer.
Math. Soc. 18, 65-72.
Winker, S.
1992. Absorption and idempotency criteria for a problem in near-
Boolean algebras. J. Algebra 153(2), 414-423.
Wooyenaka, Y.
1951. Remark on a set of postulates for distributive lattices. Proc.
Japan Acad. 27, 162-165.
1964. On postulate sets for Newman algebra and Boolean algebra. I.II.
Proc. Japan Acad. 40, 76-81; 82-87.
Yule, D.
1926. Zur Grundlegung des Klassenkalkuls. Math. Ann. 95, 446-452.
Zachik, A.I.
1974. On the axiomatization of distributive lattices (Russian). Mat.
Issled. (Kishinev) 1(31), 143-150.
Zelinka, B.
1967. D ukaz nez avislosti Birkho ova syst emu postul at u pro distribu-
tivn svazy s jednotokov ym prvkem. Casopis P est. Mat. 91, 472-477.
Zylinski, E.
1925. Some remarks concerning the theory of deduction. Fund. Math.
7, 203-209.
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Index
(xyz), 161
, 99
<, 40
< a; b; c >, 37
B(a; b), 33
C, 114
C(L), 71
D, 115
K-segment, 33, 47
K(a; b), 33
N, 114
P, 114
Pn(S), 4
U, 114
[a; b], 30
_, 1, 7
, 69
, 69
(x; y;z ), 174
, 1
r(K), 137
rsd(K), 137
!, 99
~ , 47
^, 1, 7
axb, 32
cn(I), 123
id(), 123
m(x;y; z ), 62
n-based, 16
s(x;y; z ), 47
x0, 71
Con(A), 122
L, 29
M, 49T, 29
0, 29
1, 29
Abad, M., 184
Abbott, J.C., 141, 193
absorptio-associative law, 10
absorption identity, 20
Adams, M.E., 117
Altwegg, M., 170, 174, 176
antisymmetry, 1
Arai, Y., 102, 193
arithmetical variety, 125
Avann, S.P., 189
Baker, K.A., 127
basis, 16
Becchio, D., 183
Bennett, A.A., 1, 86, 193
Beran, L., 81, 130, 131, 193
Bergman, G.M., 22
Bernstein, B.A., 74, 75, 77, 86, 94,
95, 100, 101, 103, 104, 193
Betazzi, R., 160, 176
Birkho , G., 2, 8, 9, 12, 39, 53, 58,
60, 63, 64, 69, 73, 81, 92, 117, 118,
128, 170, 176, 194
Birkho , G.D., 60, 81, 194
Blumenthal, L.M., 33, 175, 194
Boccioni, D., 187
Boicescu, V., 184
Boole, G., 72, 194
Boolean absorption, 117
Boolean algebra, 72
211
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
212 Index
Boolean function, 105
Boolean group, 95
Boolean lattice, 72
Boolean ring, 92
Bosbach, B., 102, 194
bounded lattice, 36
Bumcrot, R.J., 33, 194
Burali-Forti, C., 160, 176
Burris, S., 126, 135, 195
Byrne, L., 88, 94, 114, 195
Carloman, A.G., 84, 195
Cech, E., 163, 176
chain, 160
Chandran, V.R., 181, 182
Chen, C.C., 116, 140, 195
Cheremisin, A.I., 64, 195
Chinese lantern, 129
Church, A., 111, 113, 191, 195
Cignoli, R., 184
Clay, R.E., 176
clone, 3
closed betweenness, 164, 170
closed separation, 169
complement, 71
complementation, 72
complemented lattice, 72
complete existential theory, 74
completely independent, 74
conditional disjunction, 111
congruence, 122
congruence distributive, 125
congruence lattice, 122
conjugate identity, 139
Couturat, L., 114, 195
covering relation, 40
Croisot, R., 2, 11, 60, 65, 107, 189,
195, 196
cyclic order, 165
Dahn, B.I., 91, 195
Day, A., 139, 140, 195
De Morgan, 71
Dedekind, R., 8, 39, 53, 138, 196
de nitional equivalence, 69
Del Re, A., 74, 196Diamond, A.H., 74, 75, 189, 196
Dicker, R.M., 112, 196
Diego, A., 101, 196
di erence, 99
Dilworth variety, 140
Dilworth, R.P., 116
Dines, L.L., 103, 189, 196
disjointness relation, 115
distributive groupoid, 12
distributive lattice, 53
double negation, 71
Dra si ckov a, H., 68, 196
dual axiom, 8
dual theorem, 8
Dubreil-Jacotin, M.-L., 2, 11, 189,
196
Ellis, D.O., 58, 175, 196
equation, 19
equational class, 9
equational theory, 19
exception, 99
expression, 3
Feist, A., 91, 104, 202
Felscher, W., 3, 11, 12, 58, 197
Ferentinou-Nicolacopoulou, J., 61,
197
Ferrero Cotti, C., 187
Ferrero, G., 187
eld of sets, 93
Fig.2, 53, 54
Figallo, A., 184
Figure 1, 40
Filipoiu, A., 184
nitely de nable, 15
Fittelson, B., 91, 104, 202
ock, 144
forbidden-subalgebra theorem, 134
Frazer, W.D., 187
Freese, R., 139, 197
Frink, O., 87, 109, 145, 197
Funayama, N., 123
Gareau, A., 141
Gehman, H.M., 191
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Index 213
generalized Boolean ring, 93
Georgescu, G., 184
Gerrish, F., 74, 197
GGP term, 126
Gilmer Jr, R.W., 189
Glivenko, V., 32, 175, 197
Gmeiner, J.A., 160, 178
Goetz, A., 107, 197
Goodstein, R.L., 88, 197
Gould, M.I., 125, 197
Gr atzer, G., 23, 82, 92, 93, 116, 122,
125, 127, 135, 140, 195, 197
Grau, A.A., 106, 198
greatest lower bound, 1
groupoid, 2
Gueting, R., 101, 198
Hahn, O., 113, 198
Hammer (Iv anescu), P.L., 198
Hammer, P.L., 113
Harary, F., 176, 191
Harris, K., 91, 104, 202
Hashimoto, J., 47, 65, 198
Hasse diagram, 40
Hausdor , F., 160, 176
Hedlikov a, J., 35, 198
Henle, P., 86, 198
Higman, G., 20, 198
Hoberman, S., 104, 199
Huntingtom variety, 117
Huntington I, 74
Huntington II, 113
Huntington III, 85
Huntington IV, 85
Huntington property, 117
Huntington V, 85
Huntington VI, 86
Huntington, E.V., 74, 85{87, 100,
113, 116, 160, 162, 166, 168, 176,
190, 199
ideal, 123
idempotent, 92
identity, 3
Imai, Y., 102, 199
implication, 99implication algebra, 141
Ingraham, M.H., 189
interval, 30
irredundant, 191
Is eki, K., 62, 98, 102, 193, 199
Isobe, K., 95, 199
J onsson term, 21
J onsson, B., 139
Jezek, J., 139, 197
Jipsen, P., 139, 199
join semilattice, 1
Kalicki, J., 107
Kalman, J.A., 9, 18, 144, 199
Kalmbach, G., 129, 134, 135, 142,
145, 200
Katri~ n ak, T., 35, 37, 164, 177, 198,
200
Kelly, D., 36, 39, 48, 50, 51, 79, 134,
136{138, 200
Kelly, L.M., 47, 200
Kempe, A.B., 177, 200
Kimura, N., 8, 200
Kiss, S.A., 63, 194
Klein-Barmen, F., 2, 3, 9, 11, 197,
200
Kline, J.R., 162, 177
Kobayasi, M., 9, 201
Kolibiar, M., 33, 35, 42, 46{49, 66,
201
Kurosh, A.G., 41, 201
Lalan, V., 105, 201
Langford, C.H., 87, 201
lattice, 1
lattice betweenness, 31, 47
lattice-convex set, 33
least upper bound, 1
Lesieur, L., 2, 11, 189, 196
Lewis, C.I., 87, 201
Lihov a, J., 177
Lisovik, L.P., 84, 86, 201
Lowig, H.F.J., 62, 138, 201
Lukasiewicz-Moisil algebra, 183
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
214 Index
M3, 55
MacNeille, H.M., 114, 177, 190, 202
majority polynomial, 21
Mal'cev term, 135
Mal'cev, A.I., 135
Malliah, C., 10, 67, 89, 202
Marcisov a, T., 48, 66, 201
Martin, L.H., 36, 48, 202
Matusima, Y., 12, 60, 202
McCune, W., 16, 19, 23, 91, 104, 108,
119, 147, 202, 204
McKenzie, R.N., 13, 20, 22, 28, 119,
127, 139, 202
McKinsey, J.C.C., 104, 196, 199
McNulty, G., 137
McPhee, J.A., 165, 177
median operation, 62
meet semilattice, 1
Mendelsohn, N.S., 96, 202
Meredith, C.A., 91, 104, 144, 203
Miller, D.G., 94, 203
modular lattice, 39
Moisil, Gr.C., 86, 113, 163, 167, 177,
183, 184, 203
Montague, R., 75, 77, 203
Monteiro, A., 184
Monteiro, L., 185
Moore, E.H., 74, 189, 190, 203
Morgado, J., 98, 203
Morinaga, K., 172{174, 177
N5, 40, 55
Nakayama, T., 123
NAND, 99
Nation, J.B., 139, 140, 197
Neumann, B.H., 20, 198
Newman algebra, 81
Newman, M.H.A., 75, 81, 203
Nieminen, U.J., 112, 203
Nishig ori, N., 30, 41, 172{174, 177,
203
NOR, 99
O6, 134
O8, 134
Ohashi, S., 62, 199, 203open betweenness, 161
Ore, O., 1, 8, 39, 41, 204
ortholattice, 81, 129
orthomodular lattice, 129
Otter, 23, 91, 104, 108
Padmanabhan, R., 2, 4, 12, 16,
18{21, 23, 36, 39, 48, 50, 51, 79, 82,
87, 96, 108, 119, 127, 134, 136{138,
141, 175, 177, 181, 182, 190, 200,
202, 204
paramodulation, 24
partial order, 159
partially ordered set, 160
Peirce, C.S., 1, 53, 113, 160, 177, 204
Peirce operation, 99
Penner, P., 48, 182, 204
Pereira, R.C., 113, 205
permutable congruence(s), 124, 125
Petcu, A., 2, 4, 10, 14, 86, 183, 185,
205
Petre, G., 189, 190
Phillips, J.D., 91, 205
Pic, Gh., 68, 205
Picu, C.I., 57, 205
Pixley, A.F., 125, 205
Plo s cica, M., 35, 205
Plonka, J., 181, 182
polynomial, 3, 15
Ponticopoulos, L., 16, 42, 58, 84, 205
poset, 160
Potts, D.H., 2, 21, 205
principle of duality, 8, 39
Prior, A.N., 91, 144, 203
Problem 1, 170
Problem 64, 64
Problem 65, 60
Problem 7, 12
Prover9, 19, 28, 59, 142, 147
Quackenbush, R.W., 108, 127, 204
quasilattice, 181
re
exivity, 1
regular groupoid, 12
regular identity, 20, 181
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
Index 215
rejection, 99
Rie can, J., 42, 206
Robbins' axiom, 91
Robinson, D.F., 189, 190
Rose, H., 139, 199
Rosenbaum, I., 162, 178, 190
Rosinger, K.E., 168, 177
Rudeanu, S., 2, 9, 58, 77, 86, 87, 89,
92, 105, 107, 184, 185, 189, 190, 206
Ruedin, J., 3, 12, 206
Ruething, D., 74
Russell, B., 178
Sakai, Sh., 178
Salii, V.N., 117{119, 206
Sampathkumar, E., 74, 84, 102, 206
Sankappanavar, H.P., 126, 135, 195
Schr oder, E., 8, 53, 72, 113, 160, 178,
207
segment, 30
self-dual, 8, 37, 39
semidistributive lattice, 139
semilattice, 2
separation, 167
Shatunovski i, S.O., 160, 178
shearing identity, 40
She er function of nvariables, 113
She er stroke, 99
She er, H.M., 103, 207
Shepperd, J.A.H., 164, 169, 178
Sholander, M., 59, 62, 65, 68, 75, 95,
104, 171, 174, 178, 207
Sichler, J., 117
Sicoe, C., 102, 185, 207
signature, 71
simple Boolean function, 105
Sioson, F.M., 77, 81, 207
skew dual, 99
skew lattice, 9
Smiley, M.F., 31, 33, 47, 164, 207
Soboci nski, B., 2, 19, 42, 47, 61, 62,
66, 81, 90, 130, 132, 207
Sorkin, Yu.I., 1, 2, 8, 9, 16, 208
splitting lattice, 139
Stabler, R.E., 93, 98, 208
Stamm, E., 103, 208Stoltz, O., 160, 178
Stone, M.H., 74, 92, 102, 208
strict partial order, 159
Su, B., 160, 178
Suarez, A., 101, 196
sublattice, 40
subtraction, 144
Sudkamp, T.A., 42, 208
Sz asz, G., 11, 60, 187, 209
Tamura, S., 18, 19, 46, 61, 62, 94, 209
Tarski, A., 5, 19, 115, 127, 137, 144,
209
Tarski, J., 75, 77, 203
Taylor, J.S., 100, 101, 104, 190, 209
term, 3, 15
term function, 3
ternary majority decision, 106
ternary rejection, 106
TIT, 137
totally odered set, 160
transitivity, 1
Transue, W.R., 31, 33, 47, 164, 207
Trevisan, G., 65, 209
trivial lattice, 72
Trombetta, M., 178
Tschantz, S., 140
TUT, 137
type of an algebra, 2, 8
Vaida, D., 46, 58, 189, 190, 206, 209
Vailatti, G., 160, 178
Van Albada, P.J., 76, 209
Van de Walle, W.E., 162, 179, 191
variety, 9
Vassiliou, Ph., 64, 209
Vero , R., 23, 91, 104, 119, 202, 204,
210
very independent, 191
Vojt echovsk y, P., 91, 205
von Neumann, J., 117, 118, 128, 194
Wang, Sh.Ch., 65, 210
Whitehead, A.N., 73, 210
Whiteman, A., 110, 210
Wiener, N., 76, 210
June 19, 2008 12:16 World Scienti c Review Volume – 9in x 6in AxiomLattices
216 Index
Winker, S., 91, 210
Wolk, B., 4
Wooyenaka, Y., 60, 81, 98, 210
Wos, L., 24, 91, 104
Yooneyama, K., 160, 179
Yule, D., 116, 210Za chik, A.I., 66, 210
Zelinka, B., 60, 210
Zylinski, E., 105, 210
Copyright Notice
© Licențiada.org respectă drepturile de proprietate intelectuală și așteaptă ca toți utilizatorii să facă același lucru. Dacă consideri că un conținut de pe site încalcă drepturile tale de autor, te rugăm să trimiți o notificare DMCA.
Acest articol: 7007.tp.indd 1 6/24/08 10:25:24 AMAxioms for Lattices and Boolean Algebras This page intentionally left blankThis page intentionally left blank N E W… [621828] (ID: 621828)
Dacă considerați că acest conținut vă încalcă drepturile de autor, vă rugăm să depuneți o cerere pe pagina noastră Copyright Takedown.