2 BCK-algebras and fuzzy theories 4 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . [611998]

Byadmin

Contents
1 Introduction 2
2 BCK-algebras and fuzzy theories 4
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Fuzzy deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Cartesian product of fuzzy deductive systems . . . . . . . . . . . . . . . . . 10
2.4 BS-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Noetherian BS-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 On homomorphism of BS-algebras . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 RL-monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Fuzzy lters of RL-monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9 Implicative fuzzy lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.10 Positive implicative and boolean fuzzy lters . . . . . . . . . . . . . . . . . . 29
2.11 Fantastic fuzzy lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Fuzzy lter theory in residuated lattices 35
3.1 Filters of residuated lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Filters of residuated lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Boolean lters (implicative lters) . . . . . . . . . . . . . . . . . . . . 39
3.2.2 G- lters (positive implicative lters) . . . . . . . . . . . . . . . . . . 40
3.3 MV- lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Fuzzy lters of residuated lattices . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Fuzzy Boolean lters (fuzzy implicative lters) . . . . . . . . . . . . . . . . . 43
3.6 Fuzzy G- lters (fuzzy positive implicative lters) . . . . . . . . . . . . . . . . 44
3.7 Fuzzy MV- lters (fuzzy fantastic lters) . . . . . . . . . . . . . . . . . . . . 45
3.8 Regular lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9 Fuzzy regular lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.10 Characterizations of some particular algebras . . . . . . . . . . . . . . . . . . 51
3.11 The relations among special lters . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Positive implicative pseudo-valuations on BCK-algebras 57
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Positive implicative pseudo-valuations on BCK-algebras . . . . . . . . . . . . 58
4.3 Fuzzy lters in BCI-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Fuzzy prime lter in MV-algebra . . . . . . . . . . . . . . . . . . . . . . . . 67
1

Chapter 1
Introduction
De nition 1. A BCK algebra is a triple (A;!;1)of type(2,0) where Ais a nonempty set,
!is a binary operation on A,12Ais an element such that the following axioms are
satis ed for every x;y;z2A:
(a1)x!x= 1;
(a2)ifx!y=y!x= 1, thenx=y;
(B)x!y(y!z)!(x!z);
(C)x!(y!z) =y!(x!z);
(K)xy!x.
IfAis a BCK-algebra, then the relation xyi x!y= 1 is a partial order on A,
called natural ordering onA; 1 is the largest element of Awith respect to this ordering.
A BCK-algebra with a smallest element 0 relative to natural ordering will be called a
bounded BCK-algebra.
A non-empty subset Sof a BCK-algebra Ais called a subalgebra ofAifx!y2S,
wheneverx;y2Sand 02S.
A BCK-algebra Ahas the following properties for any x;y;z2A:
(c1)x!1 = 1;
(c2)x(x!y)!y;
(c3) ((x!y)!y)!y=x!y;
(c4) 1!x=x;
(c5) ifxythenz!xz!yandy!zx!z.
De nition 2. IfAis a BCK-algebra, a subset DofAis called a deductive system of A if
it satis es:
(a3) 12D;
2

CHAPTER 1. INTRODUCTION
(a4)ifx;x!y2D;theny2D:
For a BCK-algebra A, we denote by Ds(A)the set of all deductive systems of A. If
XA, we denote by < X > =TfD2Ds(A) :XDg. We call< X > the deductive
system generated by X.
De nition 3. LetAbe a BCK-algebra. A fuzzy set vinAis called a fuzzy deductive system
ofAif it satis es:
(a5)v(1)v(x)for allx2A;
(a6)v(y)v(x)^v(x!y)for allx;y2A.
3

Chapter 2
BCK-algebras and fuzzy theories
2.1 Preliminaries
De nition 4. A BCK algebra is a triple (A;!;1)of type(2,0) where Ais a nonempty set,
!is a binary operation on A,12Ais an element such that the following axioms are
satis ed for every x;y;z2A:
(a1)x!x= 1;
(a2)ifx!y=y!x= 1, thenx=y;
(B)x!y(y!z)!(x!z);
(C)x!(y!z) =y!(x!z);
(K)xy!x.
IfAis a BCK-algebra, then the relation xyi x!y= 1 is a partial order on A,
called natural ordering onA; 1 is the largest element of Awith respect to this ordering.
A BCK-algebra with a smallest element 0 relative to natural ordering will be called a
bounded BCK-algebra.
A non-empty subset Sof a BCK-algebra Ais called a subalgebra ofAifx!y2S,
wheneverx;y2Sand 02S.
A BCK-algebra Ahas the following properties for any x;y;z2A:
(c1)x!1 = 1;
(c2)x(x!y)!y;
(c3) ((x!y)!y)!y=x!y;
(c4) 1!x=x;
(c5) ifxythenz!xz!yandy!zx!z.
De nition 5. IfAis a BCK-algebra, a subset DofAis called a deductive system of A if
it satis es:
4

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
(a3) 12D;
(a4)ifx;x!y2D;theny2D:
For a BCK-algebra A, we denote by Ds(A)the set of all deductive systems of A. If
XA, we denote by < X > =TfD2Ds(A) :XDg. We call< X > the deductive
system generated by X.
Remark 1. The intersection of all deductive systems of a BCK-algebra Ais also a deductive
system ofA.
LetIbe a totally ordered set and fD : 2Igbe a family of deductive systems of a
BCK-algebra Asuch that for all ; 2I; > if and only if D D . Then[
2ID is a
deductive system of A.
De nition 6. LetSbe a set. A fuzzy set in Sis a function v:S![0;1].
For 2[0;1], the set
v =fx2S:v(x) g
is called a level subset of v.
A fuzzy set vin a setShas the sup property if for any subset TofS, there exists 02T
such that
v( 0) = sup
2Tv( ):
De nition 7. A fuzzy relation on any set Sis a fuzzy set v:SS![0;1].
De nition 8. Ifvis a fuzzy relation on a set Sandwis a fuzzy set in S, thenvis a fuzzy
relation on wif
v(x;y)w(x)^w(y)
for allx;y2S.
De nition 9. Letvandwbe fuzzy sets in a set S. The Cartesian product of vandwis
de ned by
(vw)(x;y) =v(x)^w(y)
for allx;y2S.
Lemma 1. Letvandwbe fuzzy sets in a set S. Then
(i)vwis a fuzzy relation on S,
(ii) (vw) =v w for all 2[0;1].
De nition 10. Ifwis a fuzzy set in a set S, the strongest fuzzy relation on Sthat is a
fuzzy relation on wisvw, given by
vw(x;y) =w(x)^w(y)
for allx;y2S.
5

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Lemma 2. For a given fuzzy set win a setS, letvwbe the strongest fuzzy relation on S.
Then for 2[0;1], we have that (vw) =w w .
De nition 11. LetAbe a BCK-algebra. A fuzzy set vinAis called a fuzzy deductive
system ofAif it satis es:
(a5)v(1)v(x)for allx2A;
(a6)v(y)v(x)^v(x!y)for allx;y2A.
2.2 Fuzzy deductive systems
Proposition 1. Letvbe a fuzzy deductive system of a BCK-algebra A. Then
(i)ifv(x!y) =v(1)thenv(x)v(y);
(ii)ifxythenv(x)v(y);
(iii)ifx!(y!z) = 1 thenv(z)v(x)^v(y).
Proof. (i) Ifv(x!y) =v(1) then
v(y)v(x)^v(x!y) =v(x)^v(1) =v(x):
(ii) Ifxythenx!y= 1. Hence using v(x!y) =v(1) we obtain v(x)v(y).
(iii) Ifx!(y!z) = 1 then xy!zand using ( ii), we obtain v(x)v(y!z).
Hence
v(z)v(y)^v(y!z)v(x)^v(y):
Theorem 1. Letvbe a fuzzy set in a BCK-algebra A. Thenvis a fuzzy deductive system
ofAif and only if for every 2[0;1], the level subset v is a deductive system of A, for all
v 6=;.
Proof. Letvbe a fuzzy deductive system of A, thenv(1)v(x) for allx2A; particularly,
v(1)v(x) for everyx2v . Hence 12v . Letx;x!y2v , Thenv(x) and
v(x!y) . It follows that
v(y)v(x)^v(x!y) ;
so thaty2v . Therefore v is a fuzzy deductive system of A.
Conversely, if v(1)v(x) for allx2A, then there exists x02Asuch thatv(1)<v(x0).
Let 0=1
2(v(1) +v(x0)). Thenv(1)< 0and 0 0< v(x0)1. Hencex02v 0, and
v 06=;. Sincev 0is a deductive system of A, therefore 12v 0, and sov(1) 0. This is a
contradiction and so v(1)v(x) for allx2A.
Now assume that v(y)v(x)^v(x!y) for allx;y2A. Then there exists x0;y02A
such that
6

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
v(y0)<v(x0)^v(x0!y0):
Let 0=v(y0) +v(x0)^v(x0!y0)
2. Thenv(y0)< 0and 0 0<v(x0)^v(x0!y0)1.
It follows that v(x0)> 0andv(x0!y0)> 0, so thatx02v 0andx0!y02v 0. This
means that v 06=;. Asv 0is a deductive system of A, we havey02v 0, and sov(y0) 0,
contradiction. This completes the proof.
De nition 12. Letvbe a fuzzy deductive system of a BCK-algebra A. The deductive system
v ; 2[0;1], are called level deductive system of v, whenv 6=;.
Theorem 2. Any deductive system of a BCK-algebra Acan be realized as a level deductive
system of some fuzzy deductive system of A.
Proof. LetDbe a deductive system of a BCK-algebra Aandvbe a fuzzy set in Ade ned
by
v(x) =(
ifx2D;
0 ifx =2D;
where is a xed number in (0 ;1). Note that 12D, so thatv(1) = v(x) for allx2A.
Letx;y2A.
Ifx2Dandx!y2D, theny2D. Thus
v(y) =v(x) =v(x!y) = ;
and so
v(y)v(x)^v(x!y):
Ifx =2Dandx!y =2D, thenv(x) =v(x!y) = 0. Hence
v(y)v(x)^v(x!y):
If exactly one of xandx!ybelongs toD, the exactly one of v(x) andv(x!y) is equal
to 0. Hence
v(y)v(x)^v(x!y):
The result above shows that v(y)v(x)^v(x!y) for allx;y2A. Therefore vis a
fuzzy deductive system of Aand obviously v =D. The proof is complete.
Theorem 3. Letvbe a fuzzy deductive system of a BCK-algebra A. Then two level deductive
systemsv 1;v 2(with 1< 2) ofvare equal if and only if there is no x2Asuch that
1v(x)< 2.
Proof. Assume that v 1=v 2for 1< 2. If there exists x2Asuch that 1v(x)< 2,
thenv 2is a proper subset of v 1. This is impossible. Conversely, suppose that there is no
x2Asuch that 1v(x)< 2. Note that 1< 2impliesv 2v 1. Ifx2v 1, then
v(x) 1, and sov(x) 2becausev(x) 2. Hencex2v 2, which says v 1v 2. Thus
v 1=v 2. This completes the proof
7

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Letvbe a fuzzy set in Aand letIm(v) denote the image of v.
Theorem 4. Letvbe a fuzzy deductive system of a BCK-algebra A. IfIm(v) =f 1; 2;:::; ng,
where 1< 2< ::: < n, then the family of deductive systems v i(i= 1;2;:::;n )consti-
tutes all the level deductive systems of v.
Proof. Let 2[0;1] and =2Im(v). If < 1, thenv 1v . Sincev 1=A, we have
v =Aandv =v 1. Assume that i< < i+1(1in1), then there is no x2A
such that v(x)< i+1. It follows from Theorem 3 that v =v i+1. This shows that
for any 2[0;1] with v(1), the level deductive system v is infv ij1ing. This
completes the proof.
Lemma 3. LetAbe a BCK-algebra and let vbe a fuzzy deductive system of A. If and
belong toIm(v)such thatv =v , then = .
Theorem 5. Letvandwbe two fuzzy deductive systems of a BCK-algebra Asuch that
vandwhave the nite images, and have the identical family of level deductive systems.
IfIm(v) =f 1; 2;:::; mgandIm(w) =f 1; 2;:::; ng, where 1> 2> :::; mand
1> 2>:::> n, then
(i)m=n;
(ii)v i=w i, fori= 1;2;:::;m ;
(iii)ifx2Asuch thatv(x) = ithenw(x) = i, fori= 1;2;:::;m .
Proof. Using Theorem 4 we have that the only level deductive systems of vandwarev i
andw i, respectively. Since vandwhave the identical family of level deductive systems, it
follows that m=n, and so (i) holds. Using again Theorem 4 we get that
fv 1;v 2;:::;v mg=fw 1;w 2;:::;w mg;
and by Theorem 3 we have
v 1v 2:::v m=A;w 1w 2:::w m=A:
Hencev i=w i, fori= 1;2;:::;m ; and (ii) holds.
Letx2Abe such that v(x) = iand letv(x) = j. Thenx2v i=w i, and so
w(x) i. Hence j i, which implies w jw i. Sincex2w j=v j, therefore
i=v(x) j. It follows that v iv j. By (ii),w i=v iv j=w j. Consequently
w i=w j, and by Lemma 3 we have i= j. Thusw(x) = i. The proof is complete.
Theorem 6. Letvandwbe two fuzzy deductive systems of a BCK-algebra Asuch thatv
andwhave the nite images, and have the identical family of level deductive systems. Then
v=wif and only if Im(v) =Im(w).
Proof. ()) Is obvious.
(() Suppose that Im(v) =Im(w) =f 1; 2;:::; ng, where 1> 2> ::: > n. Let
x1;:::;xnbe distinct elements of Asuch thatv(xi) = i(1in). By Theorem 5,
w(xi) = i;(1in). Since for any x2Athere exists some isuch thatv(x) = i, and
sox2v i=w i. Hencew(x) i, it follows that w(x)v(x). By the same argument,
we havev(x)w(x). Therefore v(x) =w(x), showing that v=w. This completes the
proof.
8

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Theorem 7. LetAbe a BCK-algebra and let vbe a fuzzy set in AwithIm(v) =f 0; 1;:::; kg,
where 0> 1>:::> k. Suppose that there exists a chain of deductive systems of A:
D0D1:::Dk=A
such thatv(Dn) = n, whereDn=DnDn1,D1=;, forn= 0;1;:::;k . Thenvis a
fuzzy deductive system of A.
Proof. Since 12D0, we havev(1) = 0v(x) for allx2A. In order to prove that v
satis es the condition v(y)v(x)^v(x!y) for allx;y2A, we divide into the following
cases :
Ifxandybelong to the same Dn, thenv(x) =v(y) = n, and so
v(y)v(x)^v(x!y):
Assume that x2Diandy2Djfor everyi6=j. Without loss of generality, we may assume
thati<j . Thenv(x) = i> j=v(y), and so
v(x)^v(x!y)v(y)<v(x):
Sincex2Di, we havex2Di. It follows that x2Dj1asij1. Now we assert that
x!y =2Dj1. In fact, if not, then x!y2Dj1andx2Dj1implyy2Dj1, which
contradicts to y2Dj=DjDj1. Hencev(x!y) j, and so
v(y)v(x)^v(x!y):
Summarizing the above results, we obtain that v(y)v(x)^v(x!y) for allx;y2A.
Thereforevis a fuzzy deductive system of A.
Theorem 8. Letvbe a fuzzy deductive system of a BCK-algebra A. IfIm(v) =f 0; 1;:::; kg
with 0> 1> ::: > k, thenDn=v n;n= 0;1;:::;k , are deductive systems of Aand
v(Dn) = n;n= 0;1;:::;k , whereDn=DnDn1andD1=;.
Proof. By Theorem 4, Dn=v n(n= 0;1;:::;k ) is a deductive system of A. Obviously
v(D0) = 0. Sincev(D1) =f 0; 1g, forx2D1we havev(x) = 1, namelyv(D1) = 1.
Repeating the above argument, we have v(Dn) = n(0nk). This completes the
proof.
Theorem 9. Ifvis a fuzzy deductive system of a BCK-algebra A, then the set
Av:=fx2Ajv(x) =v(1)g
is a deductive system of A.
Proof. Clearly 12Av. Assume that x2Avandx!y2Av. Thenv(x) =v(1) =v(x!y).
Sincevis a fuzzy deductive system of A, therefore
v(y)v(x)^v(x!y) =v(1);
whencev(y) =v(1). This means that y2Av.
9

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
De nition 13. Let 0be a real number. If m2[0;1],m will mean the positive root in
case <1. We de ne v :A![0;1]byv (x) = (v(x)) .
Theorem 10. Ifvis a fuzzy deductive system of a BCK-algebra A, thenv is also a fuzzy
deductive system of AandAv =Av.
Proof. We have that v (1) = (v(1)) (v(x)) =v (x) for allx2A. Letx;y2A.
We assert that v (y)v (x)^v (x!y). In fact, suppose that v(x)v(x!y). It
follows that v(y)v(x). Hencev (x)v (x!y) andv (x)v (y), which imply that
v (y)v (x)^v (x!y). The argument is similar if v(x)v(x!y). Finally
Av =fx2Ajv (x) =v (1)g
=fx2Aj(v(x)) = (v(1)) g
=fx2Ajv(x) =v(1)g
=Av:
2.3 Cartesian product of fuzzy deductive systems
LetAandBbe BCK-algebras and let
AB=f(x;y)jx2A;y2Bg:
We de ne an operation " !" onABby
(x;y)!(x0;y0) = (x!x0;y!y0);8(x;y);(x0;y0)2AB:
Then we can easily verify that ( AB;!(1;1)) is a BCK-algebra.
Proposition 2. LetD1andD2be deductive systems of BCK-algebras AandBrespectively.
ThenD1D2is a deductive system of AB.
Proof. The proof is obvious.
Proposition 3. For a given fuzzy set win a BCK-algebra A, letvwbe the strongest fuzzy
relation on A. Ifvwis a fuzzy deductive system of AA, thenw(x)w(1)for allx2A.
Proof. Sincevwis a fuzzy deductive system of AA, therefore
vw(x;y)vw(1;1);8(x;y)2AA:
But this means that w(x)^w(y)w(1)^w(1), which implies that w(x)w(1) for all
x2A.
Proposition 4. Ifwis a fuzzy deductive system of a BCK-algebra A, then the level deductive
system ofvware given by (vw) =w w for all 2[0;1].
Proof. Is an immediate consequence of Lemma 2
10

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Theorem 11. Letvandwbe fuzzy deductive systems of a BCK-algebra A. Thenvwis
a fuzzy deductive system of AA.
Proof. First we have that for every ( x;y)2AA,
(vw)(1;1) =v(1)^w(1)v(x)^w(y) = (vw)(x;y):
Now let (x;y);(x0;y0)2AA. Then
(vw)(x;y)^(vw)((x;y)!(x0;y0))
= (vw)(x;y)^(vw)(x!x0;y!y0)
= ((v(x)^w(y)))^(v(x!y0);w(y!y0))
= (v(x)^v(x!x0))^(w(y)^w(y!y0))
v(x0)^w(y0)
= (vw)(x0;y0):
This completes the proof.
Theorem 12. Letvandwbe fuzzy sets in a BCK-algebra Asuch thatvwis a fuzzy
deductive system of AA. Then
(i)eithervofwsatis esv(1)v(x);8x2A;
(ii)ifvsatis esv(1)v(x);8x2A, then either v(x)w(1)orw(x)w(1)for allx2A;
(iii)ifwsatis esw(1)w(x);8x2A, then either v(x)v(1)orw(x)v(1)for all
x2A;
(iv)eithervorwis a fuzzy deductive system of A.
Proof. (i) If bothvandwdo not satisfy v(1)v(x);8x2Arespectivly w(1)w(x);8×2
A, then there exists x;y2Asuch thatv(x)>v(1) andw(y)>w(1). Then
(vw)(x;y) =v(x)^w(y)>v(1)^w(1) = (vw)(1;1):
This contradicts the fact that vwis a fuzzy deductive system of AA. Hence (i) holds.
(ii) Assume that vsatis esv(1)v(x);8x2Aand letx;y2Abe such that v(x)>w(1)
andw(y)>w(1). Then
(vw)(1;1) =v(1)^w(1) =w(1):
It follows that ( vw)(x;y) =v(x)^w(y)>w(1) = (vw)(1;1), which is a contradiction.
Thus (ii) is true.
(iii) This is similar to ( ii). (iv) Since, by ( i), eithervorwsatis esv(1)v(x);8x2A
orw(1)w(x);8x2A, without loss of generality we may assume that vsatis esv(1)
v(x);8x2A. Using (ii) we have that either v(x)w(1) orw(x)w(1) for allx2A. If
v(x)w(1) for allx2A, then
(vw)(x;1) =v(x)^w(1) =v(x);8x2A:
11

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Let (x;y);(x0;y0)2AA. Sincevwis a fuzzy system of AA, byv(y)v(x)^v(x!
y);8x;y2Awe have
(vw)(x0;y0)((vw)(x;y))^((vw)((x;y)!(x0;y0)))()
= ((vw)(x;y))^((vw)(x!x0;y!y0)):
If we takey=y0= 1, then
v(x0) = (vw)(x0;1)
(vw)(x;1)^(vw)(x!x0;1!1)
= (vw)(x;1)^(vw)(x!x0;1)
= (v(x)^w(1))^(v(x!x0)^w(1))
=v(x)^v(x!x0);
showing that vsatis esv(y)v(x)^v(x!y);8x;y2A. Hencevis a fuzzy deductive
system ofA.
Now we consider the case w(x)w(1) for allx2A. Suppose that v(y)> w(1) for
somey2A. Thenv(1)v(y)> w(1). Sincew(x)w(1) for allx2A, it follows that
v(1)>w(x) for allx2A. Hence (vw)(1;x) =v(1)^w(x) =w(x) for allx2A.
Takingx=x0= 1 in (), then
w(y0) = (vw)(1;y0)
(vw)(1;y)^(vw)(1!1;y!y0)
= (vw)(1;y))^(vw)(1;y!y0)
= (v(1)^w(y))^(v(1)^w(y!y0))
=w(y)^w(y!y0);
which provides that Wsatis esv(y)v(x)^v(x!y);8x;y2A. Hencewis a fuzzy
deductive system of A, and this completes the proof.
Now we give an example to show that if vwis a fuzzy deductive system of AA, then
vandwboth need not be fuzzy deductive systems of A.
Example 1. LetAbe a BCK-algebra with jAj2and let ; 2[0;1]be such that 0 
<1. De ne fuzzy sets vandw:A![0;1]byv(x) = and
w(x) =(
; ifx= 1;
1;ifx6= 1:
for allx2A, respectively. Then (vw)(x;y) =v(x)^w(y) = for all (x;y)2AA, that
is,vw:AA![0;1]is a constant function. Hence vwis a fuzzy deductive system of
AA. Nowvis a fuzzy deductive system of A, butwis not a fuzzy deductive system of A
becausewdoes not satisfy the condition w(1)w(x);8x2A.
Theorem 13. Letwbe a fuzzy set in a BCK-algebra Aand letvwbe the strongest fuzzy
relation on A. Thenwis a fuzzy deductive system of Aif and only if vwis a fuzzy deductive
system ofAA.
12

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Proof. Assume that wis a fuzzy deductive system of A. We note that for all ( x;y)2AA,
vw(x;y) =w(x)^w(y)w(1)^w(1) =vw(1;1);
showing that vwsatis esw(1)w(x);8x2A. Let (x;y);(x0;y0)2AA. Then
vw(x;y)^vw((x;y)!(x0;y0))
=vw(x;y)^vw(x!x0;y!y0)
= (w(x)^w(y))^(w(x!x0^w(y!y0)))
= (w(x)^w(x!x0))^(w(y)^w(y!y0))
w(x0)^w(y0)
=vw(x0;y0);
which proves that vwsatis es condition vw(x;y)w(x;y)^w(x!y). Hencevwis a fuzzy
deductive system of AA.
Conversely suppose that vwis a fuzzy deductive system of AA. Then
w(x)^w(y) =vw(x;y)vw(1;1) =w(1)^w(1) =w(1)
for allx;y2A. It follows that w(x)w(1) for allx2A. For any ( x;y);(x0;y0)2AA,
we have that
w(x0)^w(y0) =vw(x0;y0)
vw(x;y)^vw((x;y)!(x0;y0))
=vw(x;y)^vw(x!x0;y!y0)
= (w(x)^w(y))^(w(x!x0)^w(y!y0))
= (w(x)^w(x!x0))^(w(y)^w(y!y0)):
In particular, if we take y=y0= 1 (respectively x=x0= 1) the
w(x0)w(x)^w(x!x0)
(respectively w(y0)w(y)^w(y!y0)):
The proof is complete.
Remark 2. Letvbe a fuzzy set in a BCK-algebra A. Thenvis a fuzzy deductive system
ofAif and only if for every 2[0;1], the level subset v is a deductive system of A, when
v 6=;.
Theorem 14. Ifvis a fuzzy deductive system of a BCK-algebra A, thenv(x) = supf 2
[0;1]jx2v gfor allx2A.
Proof. Let = supf 2[0;1]jx2v gand let">0 be given. Then
"<supf 2[0;1]jx2v g;
whence " < for some 2[0;1] such that x2v . Sincev(x) , it follows that
" < v (x), so that v(x) because"was arbitrary. We now show that v(x) .
To do this, assume that v(x) =
Thenx2v
and so
2f 2[0;1]jx2v g. Hence

supf 2[0;1]jx2v g, whencev(x) . Therefore v(x) = .
13

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
LetIbe a nonempty subset of [0 ;1]. Without loss of generality we can use Ias an index
set in the following results.
Theorem 15. LetfD j 2Igbe a collection of deductive systems of a BCK-algebra Asuch
that
(i)A=[
2ID ;
(ii) > if and only if D D for all ; 2;
We de ne a fuzzy set vinAby, for allx2A,
v(x) = supf 2Ijx2D g:
Thenvis a fuzzy deductive system of A.
Proof. For any 2[0;1], we consider the following two cases:
(1) = supf 2Ij < g
(2) 6= supf 2Ij < g
For the case (1) we know that
x2v ,x2D for all < ,x2\
< D ;
whencev( ) =\
< D , which is a deductive system of A.
Case (2) implies that there exists " > 0 such that ( "; )\I=;. We claim that
v =[
 D . Ifx2[
 D , thenx2D for some  . It follows that v(x)  , so
thatx2v . Conversely if x =2[
 D thenx =2D for all  , which implies that x =2D
for all > ", that is, if x2D then  ". Thusv(x) "and sox =2v .
Thereforev =[
 D , which is a deductive system of A. Using Remark 2, we know that v
is a fuzzy deductive system of A.
De nition 14. LetSbe a nonempty set. By an extension of fuzzy set vinSto a fuzzy set
win a setAcontaining S, we mean a fuzzy set winAsuch thatw=vinS.
Lemma 4. LetSbe a nonempty subset of a set Aand letvbe a fuzzy set in Ssuch thatv
has the sup property. If B=fB j 2Im(v)gis a collection of subsets of Asuch that
(i)[
2Im(v)B =A;
(ii) > if and only if B B for all ; 2Im(v);
14

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
(iii)v \B =v for all ; 2Im(v);  ;
thenvhas a unique extension to a fuzzy set veinAsuch that<ve> =B for all 2Im(v)
andIm(ve) =Im(v).
Remark 3. For a BCK-subalgebra Sof a bounded BCK-algebra A, ifD2Ds(S), we shall
denote byDethe deductive system of Agenerated by D.
Theorem 16. LetSbe a BCK-subalgebra of a bounded BCK-algebra Aand letvbe a fuzzy
deductive system of Ssuch thatvhas the sup property. If[
2Im(v)< v >e=Aand for all
; 2Im(v),  ,v \<v >e=v , thenvhas a unique extension to a fuzzy deductive
systemveofAsuch that<ve> =<v >efor all 2Im(v)andIm(ve) =Im(v).
Proof. Since > if and only if v v for all ; 2Im(v), the condition v \<v >e=v
implies that > if and only if <v >e<v >e. If we letB =<v >e, then by Lemma
4 we know that vhas a unique extension to a fuzzy set veinAsuch that<ve> =<v >e
for all 2Im(v) andIm(ve) =Im(v). Noticing that < ve> =< v >eis a deductive
system ofA, and using Lemma 4, we conclude that veis a fuzzy deductive system of A.
2.4 BS-algebras
De nition 15. A nonempty set Awith two binary operation "!"and""and constant
1is called an BS-algebra if Asatis es the axioms:
(a7)B(A) = (A;!;1)is a BCK-algebra;
(a8)S(A) = (A;)is a semigroup;
(a9)The operation "" is distributive over the operation "!", that is
x(y!z) = (xy)!(xz)
and
(x!y)z= (xz)!(yz)
for allx;y;z2A.
Remark 4. The multiplication xywill be used in what follows as xy.
Example 2. LetA=f1;a;b;cg. De ne "!"operation and multiplication ""by the
following tables :
! 1 a b c
11 a b c
a1 1 b c
b1 a 1 c
c1 a b 11 a b c
11 1 1 1
a1 a 1 a
b1 1 b b
c1 a b c
Using routine calculations, we can see that Ais a BS-algebra.
15

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Remark 5. In an BS-algebra A, we have 1x=x1 = 1 for anyx2A.
De nition 16. A nonempty subset Dof a semigroup S(A) = (A;)is said to be left (re-
spectively right) stable if xa2D(respectively ax2D), whenever x2S(A)anda2D.
De nition 17. A nonempty subset Dof an BS-algebra Ais called a left (respectively right)
deductive system of Aif
(a10)Dis a left (respectively right) stable subset of S(A);
(a11)for anyx;y2B(A);x!y2Dandx2Dimply thaty2D.
Remark 6. IfDis a left (respectively right) deductive system of A, then 12D. ThusDis
a deductive system of B(A).
Example 3. LetA=f1;a;b;cg. We de ne!operation and multiplication ""by the
following table:
! 1 a b c
11 a b c
a1 1 b c
b1 1 1 c
c1 1 1 11 a b c
11 1 1 1
a1 a 1 1
b1 1 b c
c1 1 c b
ThenAis a BS-algebra.
De nition 18. A fuzzy set vin a semigroup Ais called stable if v(xy)v(y)(respectively
v(xy)v(x)) for allx;y2A.
De nition 19. A fuzzy set vin an BS-algebra Ais called a fuzzy left (respectively right)
deductive system of Aif
(a12)vis a fuzzy left (respectively right) stable set in S(A);
(a13)v(x)v(x!y)^v(x)for allx;y2B(A).
Remark 7. By a (fuzzy) deductive system we shall mean a (fuzzy) left deductive system.
Example 4. Consider an BS-algebra A=f1;a;b;cgwith the following Cayley table:
! 1 a b c
11 a b c
a1 1 b c
b1 a 1 c
c1 a b 11 a b c
11 1 1 1
a1 a 1 a
b1 1 b b
c1 a b c
We de ne a fuzzy set vinAbyv(1) =v(a) = 0:7andv(b) =v(c) = 0:5. Thenvis a
fuzzy deductive system of A.
Theorem 17. Letvbe a fuzzy set in an BS-algebra A. Thenvis a fuzzy deductive system of
Aif and only if the nonempty level set v ofvis a deductive system of Afor every 2[0;1].
16

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Proof. Suppose that vis a fuzzy deductive system of A. Letx2S(A) andy2v . Then
v(y) , and that v(xy)v(y) , which implies that xy2valpha. Hencev is stable
subset ofS(A). Letx;y2B(A) be such that x!y2v andx2v . Thenv(x!y)
andv(x) . It follows that v(y)v(x!y)^v(x) , and thaty2v . Hencev is a
deductive system of A.
Conversely, assume that the nonempty level set v is a deductive system of Afor every
2[0;1]. If there exists x0;y02S(A) such that v(x0y0)< v(y0), then by taking 0=
v(x0y0) +v(y0)
2, we havev(x0y0)< 0< v(y0). It follows that y02v 0andx0y0=2valpha 0.
This is a contradiction. Therefore vis a fuzzy stable set in S(A). Suppose that v(y0)<
v(x0!y0)^v(x0) for some x0;y02X. Puttings0=v(y0) +v(x0!y0)^v(x0)
2, then
v(y0)< s 0< v(x0!y0)^v(x0), which shows that x0!y02vs0.x02vs0andy02vs0,
that is impossible. Therefore vis a fuzzy deductive system of A.
Corollary 1. LetDbe a deductive system of an BS-algebra Aand letvbe a fuzzy set of A
de ned by
v(x) =(
0ifx2D;
1otherwise
where 0> 1in[0;1]. Thenvis a fuzzy deductive system of AandU(v; 0) =D.
Proof. Notice that
U(v; 0) =8
><
>:;if 0< 1;
Dif 1< < 0;
Aif < 1
Using Theorem 17 it follows that vis a fuzzy deductive system of A. Clearly,v 0=D.
Corollary 1 suggests that any deductive system of an BS-algebra Acan be realized as a
level deductive system of some fuzzy deductive system of A. We now consider the converse
of Corollary 1.
Corollary 2. For a nonempty subset Dof an BS-algebra A, letvbe a fuzzy set in Aas
given in Corollary 1. If vis a fuzzy deductive system of A, thenDis a deductive system of
A.
Proof. Assume that vis a fuzzy deductive system of Aand letx2S(A) andy2D. Then
v(xy)v(y)tand soxy2v 0=A. HenceDis a stable subset of S(A). Letx;y2B(A)
be such that x!y2Dandx2D. It follows that v(y)v(x!y)^v(x) = 0so that
y2v 0=D. This completes the proof.
We will now show that the concept of a fuzzy deductive system of an BS-algebra is a
generalization of a deductive system.
Theorem 18. LetDbe a nonempty subset of an BS-algebra and let vbe a fuzzy set in A
such thatvis intof0;1g, so thatvis the characteristic function of D. Thenvis a fuzzy
deductive system of Aif and only if Dis a deductive system of A.
17

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Proof. The proof is obvious by using Corollary 1 and 2.
Theorem 19. Ifvis a fuzzy deductive system of an BS-algebra A, thenv(x) =supf 2
[0;1]jx2v gfor allx2A.
Proof. Let =supf 2[0;1]jx2v gand let" > 0 be given. Then " < supf 2
[0;1]jx2v g, whence " < for some 2[0;1] such that x2v . Sincev(x) ,
it follows that " < v (x), so that v(x) because"is arbitrary. Now we show that
v(x) . Assuming that v(x) =
, thenx2v
and so
2f 2[0;1]jx2v g. Hence

supf 2[0;1]jx2v g. Whencev(x) . Therefore v(x) = .
Remark 8. The intersection of all deductive systems of an BS-algebra Ais also a deductive
system ofA.
LetIbe a totally ordered set and let fD j 2Igbe a family of deductive system of an
BS-algebra Asuch that for all ; 2I, > if and only if D D . Then[
2ID is a
deductive system of A.
LetIbe a nonempty subset of [0 ;1]. Without loss of generality we can use Ias an index
set in the following results.
Theorem 20. LetfD j 2Igbe a collection of deductive systems of an BS-algebra Asuch
that
(i)A=[
2ID ;
(ii) > if and only if D D for all ; 2I.
We de ne a fuzzy set vinAby, for allx2A,
v(x) =supf 2Ijx2D g:
Thenvis a fuzzy deductive system of A.
Proof. For any 2[0;1] we consider the following two cases:
(i) = supf 2Ij < g
(ii) 6= supf 2Ij < g
For the case (i) we know that x2v is equivalent with x2D for all < which means
x2\
< D whencev =\
< D , which is a deductive system of X.
Case (ii) implies that there exists " > 0 such that ( "; )\I=;. We claim that
v =[
 D . Ifx2[
 D , thenx2D for some  . It follows that v(x)  , so
thatx2v .
Conversely if x2[
 D ,thenx2D for all  , which implies that x2D for all
 ", that is, if x =2I , then  ". Thusv(x) ", and sox =2U(v; ).
Thereforev =[
 D , which is a fuzzy deductive system of A. Therefore vis a fuzzy
deductive system of A.
18

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
De nition 20. Letvandwbe fuzzy sets of BS-algebras A1andA2respectively. The product
vwofvandwis the element of A1A2which is de ned by
(vw)(x;y) =v(x)^w(y);8(x;y)2X1X2:
Theorem 21. Ifvandware fuzzy deductive systems of BS-algebra A1andA2respectively,
thenvwis a fuzzy deductive system of A1A2.
Proof. Letx= (x1;x2),y= (y1;y2)2A1A2. Then
(vw)(xy) = (vw)((x1;x2)(y1;y2))
= (vw)(x1y1;x2y2)
=v(x1y1)^w(x2y2)
v(y1)^w(y2)
= (vw)(y1;y2)
= (vw)(y):
and
(vw)(y) = (vw)(y1;y2)
=v(y1)^w(y2)
(v(x1!y1)^v(x1))^(w(x2!y2)^w(x2))
= (v(x1!y1)^w(x2!y2))^(v(x1)^w(x2))
= (vw)(x1!y1;x2!y2)^(vw)(x1;y1)
= (vw)((x1;x2)!(y1;y2))^(vw)(x1;x2)
= (vw)(x!y)^(vw)(x):
Hencevwis a fuzzy deductive system of A1A2.
2.5 Noetherian BS-algebras
De nition 21. An BS-algebra Ais said to satisfy the ascending (descending) chain condition
( brie
y, ACC (DCC)) if for every ascending (descending) sequence D1D2:::(D1
D2:::)of deductive systems of Athere exists a natural number nsuch thatDi=Dnfor
allin.
De nition 22. An BS-algebra Ais said to be Noetherian if Asatis es the ascending chain
condition for deductive systems.
Letvbe fuzzy set in A. We note that Im(v) is a bounded subset of [0 ;1]. Hence we can
consider a sequence of elements of Im(v) is either increasing or decreasing.
Theorem 22. LetAbe an BS-algebra satisfying the descending chain condition and let v
be a fuzzy deductive system of A. If a sequence of elements of Im(v)is strictly increasing,
thenvhas a nite numbers of values.
Proof. Letf ngbe a strictly increasing sequence of element of Im(v). Then 0 1 2
:::1. De nev r=fx2Xjv(x) rg;r= 2;3;:::: Thenv ris a deductive systems of
A.
19

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Letx2v r. Thenv(x) r> tr1, which implies that x2v r1. Hencev rv r1.
Since r12Im(v), there exists xr12Asuch thatv(xr1) = r1. It follows that xr12
vr1, butxr1=2v r. Thusv r(v r1, and so we obtain a strictly descending sequence
v 1(v 2(v 3:::of deductive systems of Awhich is not terminating. This contradicts the
assumption Asatis es the descending chain condition, completing the proof.
Now we consider the converse of Theorem 22:
Theorem 23. LetAbe an BS-algebra. If every fuzzy deductive system of Ahas nite
number of values, then Asatis es the descending chain condition.
Proof. Suppose that Adoes not satisfy the descending chain condition. Then there exists a
strictly descending chain
D0)D1)D2):::
deductive systems of A. De ne a fuzzy set winAby
w(x) =8
>><
>>:n
n+ 1;ifx2DnDn+1;n= 0;1;2;:::
1; ifx21\
n=1Dn:
whereD0stands forA. We prove that wis a fuzzy deductive system of A. Letx;y2A. If
y2DnDn+1thenxy2DnsinceDnis deductive system of A, and thatv(xy)n
n+ 1=
v(y). Now, let y2n\
n=0. SinceDnis a deductive systems for any integer number n, then
n\
n=0Dnis also a deductive system. Hence xy21\
n=0Dn, and thatv(xy) = 1 =v(y). Therefore
vis a fuzzy stable subset in S(A).
Letx;y2A. Assume that x!y2DnDn+1andx2DkDk+1forn= 0;1;2;:::;
k= 0;1;2;:::. Without loss of generality, we may assume that nk. Then clearly x2Dn.
SinceDnis a deductive system, we have y2An. Hencew(y)n
n+ 1=w(x!y)^w(x).
Ifx!y,x21\
n=0Dnandy21\
n=0Dn. Thusw(y) = 1 =w(x!y)^w(x). Ifx!y =21\
n=0Dn
andy21\
n=0Dn, then there exists k2N, such that x!y2Dk!Dk+1. It follows that
y2Dk, so thatw(y)k
k+ 1=w(x!y)^w(x). Finally, assume that x!y21\
n=0An
andx =21\
n=0An. Thenx2ArAr+1for somer2N. It follows that y2Dr, and hence
w(y)k
k+ 1=w(x!y)^w(x). Consequently, we nd that wis a fuzzy deductive
system and whas in nite number of di erent values. This is a contradiction and the proof
is complete.
20

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Theorem 24. For any BS-algebra A, the following are equivalent:
(i)Xis Noetherian;
(ii)The set of values of any fuzzy deductive system on Ais a well-ordered subset of [0;1].
Proof. Suppose that vis fuzzy deductive system whose set of values is not a well-ordered
subset of [0 ;1]. Then there exists a strictly decreasing sequence f ngsuch thatv(xn) = n.
LetB=fx2Ajv(x)2 ng. ThenB1(B2(B3:::is a strictly ascending chain of
deductive systems of X, contradicting the assumption that Xis Noetherian.
Assume that the condition is satis ed and Ais not Noetherian. Then there exists a strictly
ascending chain D1(D2(D3(:::of deductive system of A. Suppose that D=[
n2NDn.
ThenDis a deductive system of A.
De ne a fuzzy set winAby:
w(x) =8
<
:0;ifx =2Dn
1
k;wherek=minfn2Njx2Dng:
We claim that wis a fuzzy deductive system of A. Letx;y2A. Ify2DnDn1for
n= 2;3;:::, thenxy2Dn. It follows that w(xy)1
n=w(y). Ify =2Dn, thenw(y) = 0,
and thatw(xy)w(y). Therefore vis a fuzzy stance set in S(A). Now, let x;y2A. If
x!y,x2DnDn1forn= 2;3;:::, theny2Dn. It follows that
w(y)1
n=w(x!y)^w(y)
Assume that x;y2Dnandx2DnDmfor allm<n . SinceDnis a deductive system,
thereforey2Dn. Hencew(y)w(x!y)^w(x).
Similarly for the case x!y2DnDmandx2Dn, we havew(y)w(x!y)^w(x).
Thuswis a fuzzy deductive system of A. Since the chain is not terminating, whas a strictly
descending sequence of values. This contradicts the assumption that the value set of any
fuzzy deductive system is well-ordered. Hence Ais Noetherian.
We note that a set is well-ordered if and only if it does not contain any in nite descending
sequence.
Theorem 25. LetS=f 1; 2;:::g[f 0g, wheref ngis a strictly decreasing sequence
in(0;1). Then an BS-algebra Ais Noetherian if and only if for each fuzzy deductive sys-
temvofA,Im(v)Simplies that there exists a positive integer n0such thatIm(v)
f 1; 2;:::; n0g[f 0g.
Proof. IfAis a Noetherian BS-algebra, then we know from Theorem 24 that Im(v) is a
well-ordered subset of [0 ;1] and so the condition is necessary.
Conversely, assume that the condition is satis ed. Suppose that Ais not Noetherian.
Then there exists a strictly ascending chain of deductive systems D1(D2(D3:::.
21

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
De ne a fuzzy set vinAby
v(x) =8
>>><
>>>: 1;ifx2D1
n;ifx2DnDn1;n= 2;3;:::
0; ifx2A1[
n=1Dn:
Ify2ADn, thenv(y) = 0. Hence v(xy)v(y). Ify2D1,thenxy2D1, and
sov(xy) = 1=v(y). Ify2DnDn1forn= 2;3;:::, thenxy2Dn, and hence
v(xy) n=v(y). Therefore vis a fuzzy stable subset in S(A).
Now letx;y2A. If eitherx!yorAbelong toADn, then either v(x!y) orv(x)
is equal to 0. Hence v(y)w(x!y)^w(x). Ifx!y,x2D1, theny2Dn. Hence
v(y)tn=v(x!y)^v(y). Assume that x!y2D1andx2DnDn1forn= 2;3;:::,
theny2Dnand hencev(y) n= 1^ n=v(x!y)^v(x). Similarly for y2D1and
x!y2DnDn1;n= 2;3;:::, we obtain v(y) n=v(x!y)^v(x). Hencevis a
fuzzy deductive system of A. This contradicts our assumption.
2.6 On homomorphism of BS-algebras
De nition 23. A mapping f:A!Bof BS-algebras is called a homomorphism if
(a14)f(x!y) =f(x)!f(y)for allx;y2B(A);
(a15)f(xy) =f(x)f(y)for allx;y2S(A).
Remark 9. Iff:A!Bis a homomorphism of BS-algebra, then f(1) = 1 .
Letf:A!Bbe a homomorphism of BS-algebras. For any fuzzy set vinYwe de ne a
setvfinAbyvf(x) =v(f(x)) for allx2A.
Theorem 26. Letf:A!Bbe a homomorphism of BS-algebra. If vis a fuzzy deductive
system ofB, thenvfis a fuzzy deductive system of A.
Proof. Assume that vis a fuzzy deductive system of B, thenvf(xy) =v(f(xy)) =v(f(x)f(y))
v(f(y)) =vf(y) andvf(y) =v(f(y))v(f(x)!f(y))^v(f(x)) =v(f(x!y))^v(f(x)) =
vf(x!y)^vf(x). Hencevfis a fuzzy deductive system of A.
If we strengthen the condition f, then the converse of Theorem 26 is obtained as follows.
Theorem 27. Letf:A!Bbe an epimorphism of BS-algebras. If vfis a fuzzy deductive
system ofA, thenvis a fuzzy deductive system of B.
Proof. For anyx;y2B, there exists a;b2Asuch thatf(a) =xandf(b) =y. Then
v(xy) =v(f(a)f(b)) =v(f(ab))
=vf(ab)vf(b) =v(f(b))
=v(y)
22

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
and
v(y) =v(f(b))
=vf(b)vf(a!b)^vf(a)
=v(f(a!b))^v(f(a))g
=v(f(a)!f(a))^v(f(b))
=v(x!y)^v(x):
Hencevis a fuzzy deductive system of B.
2.7 RL-monoids
De nition 24. A bounded commutative RL-monoid is an algebra L= (L; ;_;^;!;0;1)
of type<2;2;2;2;0;0>satisfying the following conditions:
(a16) (L; ;1)is a commutative monoid;
(a17) (L;_;^;0;1)is a bounded lattice;
(a18)x yzif and only if xy!zfor anyx;y;z2L;
(a19)x (x!y) =x^yfor anyx;y2L.
In the sequel, by an RL-monoid we will mean a bounded commutative RL-monoid. On
any RL-monoid Lwe de ne a unary operation negation " " byx:=x!0 for anyx2L.
Remark 10. In fact, bounded commutative RL-monoids can be also recognized as commu-
tative residuated lattices satisfying the divisibility condition or as divisible integral residuated
commutative l-monoids or as bounded integral commutative generalized BL-algebras.
The above mentioned algebras can be characterized in the class of all RL-monoids as
follows: An RL-monoid Lis :
(a)BL-algebra if and only if Lsatis es the identity of pre-linearity ( x!y)_(y!x) = 1;
(b)an MV-algebra if and only if Lful ls the double negation law x=x;
(c)a Heyting algebra if and only if the operations " " is idempotent.
When doing calculations, we will use the following list of basics rules for bounded RL-
monoids.
Lemma 5. In any bounded commutative RL-monoids Lwe have for any x;y;z2L:
(c6) 1!x=x,
(c7)x yx^y,
(c8)xy!x
(c9) (x y)!z=x!(y!z) =y!(x!z),
23

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
(c10) (x_y)!z= (x!z)^(y!z),
(c11)x!(y_z) = (x!y)^(x!z),
(c12)xx,x=x,
(c13)xy)yx,
(c14) (x y)=y!x=y!x=x!y=x!y,
(c15)x!yy!x,
(c16)x_y((x!y)!y)^((y!x)!x).
A non-empty subset Fof an RL-monoid Lis called a lter ofLif
(a20)x;y2Fimplyx y2F;
(a21)x2F;y2L;xyimplyy2F.
A subsetDof an RL-monoid Lis called a deductive system ofLif
(a22) 12D;
(a23)x2D;x!y2Dimplyy2D.
Proposition 5. LetHbe a non-empty subset of an RL-monoid L. ThenHis a lter of L
if and only if His a deductive system of L.
Filters of commutative RL-monoids are exactly the kernels of their congruences. If Fis
a lter ofL, thenFis the kernel of a unique congruence ( F) such that < x;y >2(F)
if and only if ( x!y)^(y!x)2Ffor anyx;y2L. Hence we will consider quotient
RL-monoids L=F of RL-monoids Lby their lters F.
2.8 Fuzzy lters of RL-monoids
Let [0;1] be the closed unit interval of reals and let L6=;be a set. Recall that a fuzzy set
inLis any function v:L![0;1].
A fuzzy set vin an RL-monoid Lis called a fuzzy lter ofLif anyx;y2Lsatisfy:
(a24)v(x y)v(x)^v(y);
(a25)xy)v(x)v(y).
By (a25), it follows immediately that
(a26)v(1)v(x) for every x2L.
Lemma 6. Letvbe a fuzzy lter of an RL-monoid L. Then for any x;y2Lwe hasve
(i)v(x_y)v(x)^v(y);
24

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
(ii)v(x^y) =v(x)^v(y);
(iii)v(x y) =v(x)^v(y).
Proof. For anyx;y2Lwe havex yx^yx_y. Then (f2) and (f1) imply
v(x_y)v(x y)v(x)^v(y). Sincex yx^yx;y, it follows by ( f1) and (f2)
thatv(x)^v(y)v(x y)v(x^y)v(x)^v(y).
Theorem 28. A fuzzy set vin an RL-monoid Lis a fuzzy lter of Lif and only if it satis es
(f1)and
(f4)v(x_y)v(x)for anyx;y2L.
Proof. Ifvis a fuzzy lter of an RL-monoid Lthenxx_yimpliesv(x)v(x_y).
Conversely, if vsatis es (f1) and (f4) andxy, thenv(y) =v(x_y)v(x). Hencev
is a fuzzy lter of L.
Theorem 29. Letvbe a fuzzy set in an RL-monoid L. Then the following conditions are
equivalent.
(i)vis a fuzzy lter of L;
(ii)vsatis es (f3)and for all x;y2L,v(y)v(x)^v(x!y).
()v(y)v(x)^v(x!y).
Proof. (i))(ii): Letvbe a fuzzy lter of Land letx;y2L. Then, by Lemma 6 (iii),
v(y)v(x^y) =v((x!y) x) =v(x!y)^v(x). Hencevsatis es the condition (2).
(ii))(i): Letvbe a fuzzy set in Lsatisfying (f3) and (). Letx;y2L,xy. Then
x!y= 1. Thusv(y)v(x)^v(1) =v(x), hence (f2) holds.
Further, since xy^(x y), by () and (f2) we getv(x y)v(y)^v(y!(x y))
v(y)^v(x). Therefore ( f1) is also satis ed and hence vis a fuzzy lter of L.
LetFbe a subset of Land let ; 2[0;1] be such that > . De ne a fuzzy subset
vF( ; ) inLby
vF( ; )(x) :=(
; ifx2F
; otherwise:
In particular, vF(1;0) is the characteristic function FofF. We will use the notation vF
instead ofvF( ; ) for every ; 2[0;1]; > .
Letvbe a fuzzy set in Land let 2[0;1]. The set
v :=fx2L:v(x) g
is called the level subset of vdetermined by .
Kondo [52] and Dudek [19] formulated an proved the so-called Transfer Principle (TP)
which can be used to any (general) algebra of any type.
Transfer Principle. A fuzzy set de ned in a (general) algebra Ahas a propertyPif
and only if all non-empty level subsets U(; )have the property P.
25

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Theorem 30. LetFbe a nonempty subset of an RL-monoid L. Then the fuzzy set vFis a
fuzzy lter of Lif only ifFis a lter of L.
LetFbe a fuzzy set in an RL-monoid L. Denote by Lvthe set
Lv:=fx2L:v(x) =v(1)g:
Note thatLv=U(v;v(1)), hence Lvis a special case of level subset of L.
Theorem 31. Ifvis a fuzzy lter of an RL-monoid L, thenLvis a lter of L.
Proof. Letvbe a fuzzy lter of L. Letx;y2Lv, i.e.v(x) =v(1) =v(y). Thenv(x y)
v(x)^v(y) =v(1), hencev(x y) =v(1), thusx y2Lv.
Further, let x2Lv;y2Landxy. Thenv(1) =v(x)v(y), hencev(y) =v(1) and
thereforey2Lv.
That means Lvis a lter of L.
The converse implication to that from Theorem 31 is not true in general, not even for
pseudo MV-algebras, as was shown in Dymek.
Theorem 32. Letvbe a fuzzy set in an RL-monoid L. Thenvis a fuzzy lter of Lif and
only if its level subset v is a lter of Lofv =;for each 2[0;1].
Proof. If follows from the Transfer Principle.
Theorem 33. Letvbe a fuzzy subset in an RL-monoid L. Then the following conditions
are equivalent
(i)vis a fuzzy lter of L;
(ii)8x;y;z2L;x!(y!z) = 1)v(z)v(x)^v(y).
Proof. (i))(ii): Letvbe a fuzzy lter of L. Letx;y;z2Landx!(y!z) = 1. Then
by Theorem 29, v(y!z)v(x)^v(x!(y!z)) =v(x)^v(1) =v(x).
Moreover, also by Theorem 29, v(z)v(y)^v(y!z), hence we obtain v(z)v(y)^v(x).
(ii))(i): Let a fuzzy set vinLsatisfy the condition (2). Let x;y2L. Sincex!(x!
1) = 1, we have v(1)v(x)^v(x) =v(x), hence (f3) is satis ed.
Further, since ( x!y)!(x!y) = 1 we get v(y)v(x!y)^v(x), thusvsatis es
(), which means, by Theorem 29, that vis a fuzzy lter of L.
Corollary 3. A fuzzy set vin an RL-monoid Lis a fuzzy lter of Lif and only if for all
x;y;z2L,x yzimpliesv(z)v(x)^v(y).
2.9 Implicative fuzzy lters
LetLbe an RL-monoid and Fa subset of L. ThenFis called an implicative lter ofLif
(i) 12F;
(ii)x!(y!z)2F;x!y2Fimplyx!z2Ffor anyx;y;z2L.
26

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Remark 11. We know that every implicative lter is a lter of L.
A fuzzy set vin an RL-monoid Lis called an implicative fuzzy lter ofLif for any
x;y;z2L
(i)v(1)v(x);
(ii)v(x!(y!z))^v(x!y)v(x!z).
Proposition 6. Every implicative fuzzy lter of an RL-monoid Lis a fuzzy lter of L.
Proof. Letvbe an implicative fuzzy lter of L. Let 2[0;1] be such that U(v; )6=;.
Then for any x2v we havev(1)v(x), thus 12v .
Letx;x!y2v , i.e.v(x);v(x!y) . Thenv(1!x);v(1!(x!y)) , hence
v(1!(x!y))^v(1!x) , thus by (2), v(1!y) . That means v(y) , and
thereforey2v . Hence by Theorem 32, vis a fuzzy lter of L.
Theorem 34. A lterFof an RL-monoid Lis implicative if and only if vFis an implicative
fuzzy lter of L.
Proof. It follows from the Transfer Principle.
Theorem 35. LetFbe a lter of an RL-monoid L. Then the following conditions are
equivalent
(i)F is an implicative lter of L;
(ii)y!(y!x)2Fimpliesy!x2Ffor anyx;y2L;
(iii)z!(y!x)2Fimplies (z!y)!(z!x)2Ffor anyx;y;z2L;
(iv)z!(y!(y!x))2Fandz2Fimplyy!x2Ffor anyx;y;z2L;
(v)x!(x x)2Ffor anyx2L.
Theorem 36. LetFbe a lter of an Rl-monoid L. Then the following conditions are
equivalent.
(i)vFis an implicative fuzzy lter of L;
(ii)vF(y!(y!x))vF(y!x)for anyx;y2L;
(iii)vF(z!(y!x))vF((z!y)!(z!x))for anyx;y;x2L;
(iv)vF(z!(y!(y!x)))^vF(z)vF(y!x)for anyx;y;z2L;
(v)vF(x!(x x)) =vF(1).
27

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Proof. (i),(ii): LetvFbe an implicative fuzzy lter of L. Then by Theorem 34, Fis an
implicative lter of L, and hence by Theorem 35, y!(y!x)2Fimpliesy!x2Ffor
anyx;y2L. Letx;y2Land letvF(y!(y!x)) = . ThenvF(y!x) = , and thusvF
satis es the condition ( ii).
Conversely, let vFsatisfy (ii). Letx;y2Landy!(y!x)2F. ThenvF(y!(y!
x)) = , hence also vF(y!x) = , that means y!x2F. Therefore, by Theorem 35, F
is an implicative fuzzy lter of L.
The proofs of the equivalences ( i),(iii);(i),(iv) and (i),(v) are analogous.
Theorem 37. Letvbe a fuzzy lter of an RL-monoid L. Thenvis an implicative fuzzy
lter ofLif and only if v is an implicative lter for any 2[0;1]such thatv 6=;.
Proof. It follows from the Transfer Principle.
As a consequence we obtain the following theorem.
Theorem 38. Ifvis a fuzzy lter of an RL-monoid L, thenvis an implicative fuzzy lter
ofLif and only if v satis es any of conditions (ii)(v)of Theorem 35 for each 2[0;1]
such thatv 6=;.
Theorem 39. IfFis a lter of an RL-monoid L, thenFis aan implicative lter if and
only if the quotient RL-monoid L=F is a Heyting algebra.
The following Theorem follows from Theorems 38 and 39.
Theorem 40. Ifvis a fuzzy lter of an RL-monoid L, thenvis an implicative fuzzy lter
ofLif and only if the quotient RL-monoid L=v is Heyting algebra for any 2[0;1]such
thatv 6=;.
Theorem 41. LetLbe an RL-monoid. Then the following conditions are equivalent.
(i)Lis a Heyting algebra;
(ii)Every lter of Lis implicative;
(iii)f1gis an implicative lter of L.
Theorem 42. LetLbe an RL-monoid. Then the following conditions are equivalent.
(i)Lis a Heyting algebra;
(ii)Every fuzzy lter of Lis implicative;
(ii)Every fuzzy lter vofLsuch thatv(1) = 1 is implicative;
(iv)f1gis an implicative fuzzy lter of L.
Proof. (i))(ii): LetLbe a Heyting algebra and va fuzzy lter of L. If 2[0;1] and
v 6=;, thenv is, by Theorem 41, an implicative lter of L. Hence, by Theorem 37, vis an
implicative fuzzy lter of L.
(ii))(iii);(iii))(iv) are obvious.
(iv))(i): If the fuzzy lter f1g=vf1g(1;0) is implicative, then by Theorem 34, f1gis
an implicative lter of L, and hence by Theorem 41, Lis a Heyting algebra.
28

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
2.10 Positive implicative and boolean fuzzy lters
LetLbe an RL-monoid and Fa subset of L. ThenFis called a positive implicative lter
ofLif
(i) 12F;
(ii)x!((y!z)!y)2Fandx2Fimplyy2Ffor anyx;y;z2L.
Remark 12. Every positive implicative lter of Lis a lter of L.
A fuzzy set vin an RL-monoid Lis called a positive implicative fuzzy lter ofLif for any
x;y;z2L
(i)v(1)v(x);
(ii)v(x!((y!z)!y))^v(x)v(y).
Proposition 7. Every positive implicative fuzzy lter of an RL-monoid Lis a fuzzy lter
ofL.
Proof. Letvb a positive implicative fuzzy lter of L, 2[0;1] and 6=;. Then 12v .
Further, let x;x!y2v , i.e.v(x);v(x!y) . Thenv(x!((y!1)!y)) =
v(x!(1!y)) =v(x!y), hencev(x!((y!1)!y)^v(x) and thus by (3),
v(y) . Therefore y2v . That means, by Theorem 32, vis a fuzzy lter of L.
Theorem 43. A lterFof an RL-monoid Lis positive implicative if and only if vFis a
positive implicative fuzzy lter of L.
Proof. LetFbe a lter of L. Let us suppose that Fis positive implicative. Let vF(x!
((y!z)!y)^vF(x) = . ThenvF(x!((y!z)!y)) = =vF(x), thusx!((y!
z)!y);x2F, and hence y2F, that means vF(y) = . Therefore we get that vFis a
positive implicative fuzzy lter of L.
Conversely, let vFbe a positive implicative fuzzy lter of L. Letx!((y!z)!y)2F
andx2F. ThenvF(x!((y!z)!y)) = =vF(x), hencevF(xy) = and soy2F.
That means Fis a positive implicative lter of L.
Theorem 44. LetFbe a lter of an RL-monoid L. Then the following conditions are
equivalent.
(i)Fis a positive implicative lter of L;
(ii) (x!y)!x2Fimpliesx2Ffor anyx;y2L;
(iii) (x!x)!x2Ffor anyx2L.
Theorem 45. LetFbe a lter of an RL-monoid L. Then the following conditions are
equivalent.
(i)vFis a positive implicative fuzzy lter of L;
29

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
(ii)vF((x!y)!x)vF(x)for anyx;y2L;
(iii)vF((x!x)!x) =vF(1).
Theorem 46. Letvbe a fuzzy lter of an RL-monoid L. Thenvis positive implicative
fuzzy lter of Lif and only if v is a positive implicative lter of Lfor every 2[0;1]such
thatv is a positive implicative lter of Lfor every 2[0;1]such thatv 6=;.
Proof. Let us suppose that vis a fuzzy lter of L. Letvbe positive implicative, 2[0;1],
v 6=;,x;y;z2Landx!((y!z)!y)2v ,x2v . Thenv(x!((y!z)!
y));v(x) , hencev(x!((y!z)!y))^v(x) . Sincev(y)v(x!((y!z)!
y))^v(x), we gety2v . Therefore the lter v is positive implicative.
Conversely, let vbe such that v is a positive implicative lter for any 2[0;1] such
thatv 6=;. Ifx;y;z2L, thenx!((y!z)!y);v(x!((y!z)!y)^x), thus also
y2v(x!((y!z)!y)^x), hencev(y)v((x!((y!z)!y))^x) =v(x!((y!
z)!y))^v(x). That means vis a positive implicative fuzzy lter.
Theorem 47. Every positive implicative fuzzy lter of an RL-monoid Lis implicative.
Proof. Letvbe a positive implicative fuzzy lter of L. Then by Theorem 46. If 2[0;1] is
such thatv 6=;thenv is a positive implicative lter of L. Hence,v is also an implicative
lter ofL. Therefore, by Theorem 37, vis an implicative fuzzy lter of L.
Theorem 48. LetFbe an implicative lter of an RL-monoid L. Then the following condi-
tions are equivalent.
(i)F is a positive implicative lter of L;
(ii) (x!y)!y2Fimplies (y!x)!x2Ffor anyx;y2F.
Theorem 49. LetFbe an implicative lter of an RL-monoid L. Then the following condi-
tions are equivalent.
(i)vFis a positive implicative lter of L;
(ii)vF((x!y)!y) =vF((y!x)!x)for anyx;y2F.
Proof. (i))(ii): LetvFbe a positive implicative fuzzy lter of L. Then by Theorem 43, F
is a positive implicative fuzzy lter of Lhence (x!y)!y2Fimplies (y!x)!x2F
for anyx;y2L. LetvF((x!y)!y) = . Then also vF((y!x)!x) = , and thusvf
satis es (b).
(ii))(i): LetvFsatisfy (b) and letvF(x!((y!z)!y)) = =vF(x). Then
x!((y!z)!y);x2F, and hence also ( y!z)!y2F. Further, ( y!z)!y(y!
z)!((y!z)!z), therefore ( y!z)!((y!z)!z)2F. SinceFis an implicative
lter ofL, by Theorem 35 we get ( y!z)!z2F, and consequently by Theorem 48, Fis
a positive implicative fuzzy lter of L. Therefore by Theorem 43, vFis a positive implicative
fuzzy lter of L.
Theorem 50. LetLbe an RL-monoid. Then the following conditions are equivalent.
30

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
(i)f1gis a positive implicative lter;
(ii)Every lter of Lis positive implicative;
(iii)Lis a Boolean algebra.
Theorem 51. LetLbe an RL-monoid. Then the following conditions are equivalent.
(i)Lis a Boolean algebra;
(ii)Every fuzzy lter of Lis positive implicative;
(iii)Every fuzzy lter vofLsuch thatv(1) = 1 is positive implicative;
(iv)f1gis a positive implicative fuzzy lter of L.
Proof. (i))(ii): LetLbe a Boolean algebra and va fuzzy lter of L. Let 2[0;1] be such
thatv :;. Then by Theorem 50, v is a positive implicative lter of L. Hence by Theorem
46 we obtain that vis a positive implicative fuzzy lter of L.
The equivalences ( ii))(iii);(iii))(iv) are obvious.
(iv)(i): Letf1g=vf1g(1;0) be a positive implicative fuzzy lter of L. Then by
Theorem 43 we get that f1gis a positive implicative lter of L, and therefore by Theorem
50,Lis a Boolean algebra.
A lterFof an RL-monoid Lis called a Boolean lter ofL, if for any x2L,x_x2F.
A fuzzy lter vof an RL-monoid Lis called a Boolean fuzzy lter ofL, if for any
x2L;v(x_x) =v(1).
Theorem 52. A lter of an RL-monoid Lis Boolean if and only if vFis a Boolean fuzzy
lter ofL.
Proof. LetFbe a Boolean lter of Land letx2L. ThenvF(v_x) = =vF(1), hence
vFis a Boolean fuzzy lter of L.
Conversely, let vFbe a Boolean fuzzy lter of Land letx2F. ThenvF(x_x) =vF(1) =
, thenx_x2F, that means Fis a Boolean lter of L.
Theorem 53. Letvbe a fuzzy lter of an RL-monoid L. Then the following conditions are
equivalent.
(i)vis a Boolean fuzzy lter of L;
(ii)If 2[0;1]is such that v 6=;, thenv is a Boolean lter of L;
(iii)Lv=U(v;v(1)) is a Boolean lter of L.
Proof. It follows from the Transfer Principle.
Theorem 54. Letvbe a fuzzy lter of an Rl-monoid L. Thenvis Boolean if and only if
the quotient RL-monoid L=v is a Boolean algebra for any 2[0;1]such thatv 6=;.
31

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Proof. In his work, Rachin _ unek proved that a lter Fof an RL-monoid Lis Boolean if
and only if L=F is a Boolean algebra. Hence the assertion is a corollary of the preceding
theorem.
Theorem 55. A lterFof an RL-monoid Lis positive implicative if and only if Fis a
Boolean lter.
As a consequence of Theorems 43, 52 and 55 we have:
Theorem 56. IfFis a lter of an RL-monoid Lthen for the fuzzy lter vFthe following
conditions are equivalent.
(i)vFis a positive implicative fuzzy lter of L;
(ii)vFis a Boolean fuzzy lter of L.
Analogously, from Theorems 46, 53 and 55 we get:
Theorem 57. Letvbe a fuzzy lter of an RL-monoid L. Then the following conditions are
equivalent.
(i)If 2[0;1]is such that v 6=;, thenv is a positive implicative lter of L;
(ii)If 2[0;1]is such that v 6=;, thenv is a Boolean lter of L.
Remark 13. Theorems 40 and 57 give an alternative proof of Theorem 47.
2.11 Fantastic fuzzy lters
LetLbe an RL-monoid and Fa subset of L. ThenFis called a fantastic lter ofLif
(i) 12F;
(ii)z!(y!x)2Fandz2Fimply ((x!y)!y)!x2Ffor anyx;y;z2L.
A fuzzy subset vin an RL-monoid Lis called a fantastic fuzzy lter ofLif for any
x;y;z2F,
(i)v(1)v(x);
(iv)v(z!(y!x))^v(z)v(((x!y)!y)!x).
Proposition 8. Every fantastic fuzzy lter of an RL-monoid Lis a fuzzy lter of L.
Proof. Letvbe a fantastic fuzzy lter of L. Let 2[0;1] andv 6=;. Then 12v . Let
x;x!y2v , which means, v(x);v(x!y) . Thenv(x!(1!y)) =v(x!y) ,
hencev(x!(1!y))^v(x) , thus, byv(z!(y!x))^v(z)v(((x!y)!y)!x),
v(y) =v(1!y) =v(((y!1)!1)!y)v(x!(1!y))^v(x) , and soy2v .
Therefore by Theorem 32, vis a fuzzy lter of L.
32

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Theorem 58. A lterFof an RL-monoid Lis fantastic if and only if vFis a fantastic
fuzzy lter of L.
Proof. LetFbe a lter of L. Let us suppose that F is fantastic. Let vF(z!(y!
x))^vF(z) = . ThenvF(z!(y!x)) = =vF(z), thusz!(y!x)2F;z2F, and
hence ((x!y)!y)!x2F, that means v(((x!y)!y)!x) = . Therefore we get
thatvFis a fantastic fuzzy lter of L.
Conversely, let vFbe a fantastic fuzzy lter of L. Letz!(y!x)2Fandz2F.
ThenvF(z!(y!x)) = =vF(z) hencevF(((x!y)!y)!x) = , and therefore
((x!y)!y)!x2F. That means, Fis a fantastic lter of L.
Theorem 59. LetFbe a lter of an RL-monoid L. Then the following conditions are
equivalent.
(i)Fis a fantastic lter of L;
(ii)y!x2Fimplies ((x!y)!y)!x2Ffor everyx;y2L;
(iii)x!x2Ffor everyx2L;
(iv)x!z2Fandy!z2Fimplies ((x!y)!y)!z2Ffor everyx;y;z2L.
Theorem 60. LetFbe a lter of an RL-monoid L. Then the following conditions are
equivalent,
(i)vFis a fantastic fuzzy lter of L;
(ii)vF(y!x)vF(((x!y)!y)!x)for anyx;y2L;
(iii)vF(x!x) =vF(1)for anyx2L;
(iv)vF(x!z)^vF(y!z)vF(((x!y)!y)!z)for anyx;y;z2L.
Proof. The proof is analogous to that from Theorem 36.
Theorem 61. Letvbe a fuzzy lter of an RL-monoid L. Thenvis a fantastic fuzzy lter
ofLif and only if v is a fantastic lter of Lfor every 2[0;1]such thatv 6=;.
Proof. It follows from the Transfer Principle.
As a consequence we obtain the following theorem.
Theorem 62. Letvbe a fuzzy lter of an RL-monoid L. Thenvis a fantastic fuzzy lter of
Lif and only if v satis es each of conditions (b)(d)of Theorem 59 a for every 2[0;1]
such thatv 6=;.
Theorem 63. Every positive implicative fuzzy lter of an RL-monoid is fantastic.
Proof. Ifvis a positive implicative fuzzy lter of L, then by Theorem 46, v is a positive
implicative lter of Lfor everyv 6=;. Hence,v is also a fantastic lter of L, and hence,
by Theorem 62, vis a fantastic lter of L.
33

CHAPTER 2. BCK-ALGEBRAS AND FUZZY THEORIES
Theorem 64. A lterFof an RL-monoid Lis fantastic if and only if L=F is an MV-algebra.
Theorem 65. Ifvis a fuzzy lter of an RL-monoid L, thenvis a fantastic fuzzy lter of L
if and only if the quotient RL-monoid L=v is an MV-algebra for every 2[0;1]such that
v 6=;.
Proof. The proof follows from Theorems 61 and 64.
Theorem 66. LetLbe an RL-monoid. Then the following conditions are equivalent.
(i)Lis an MV-algebra;
(ii)Every lter of Lis fantastic;
(iii)f1gis a fantastic lter of L.
Theorem 67. LetLbe an Rl-monoid. Then the following conditions are equivalent:
(i)Lis an MV-algebra;
(ii)Every fuzzy lter of Lis fantastic;
(iii)Every fuzzy lter vofLsuch thatv(1) = 0 is fantastic;
(iv)f1gis a fantastic fuzzy lter of L.
Proof. (i))(ii) Letvbe a fuzzy lter of an MV-algebra L. Let 2[0;1] be such that
U(v; )6=;. Then by Theorem 66, U(v; ) is a fantastic lter of L, hence by Theorem 61,
vis a fantastic fuzzy lter of L.
The implications ( ii))(iii) and (iii))(iv) are obvious.
(iv))(i) Letf1g=vf1g(1; 0) be a fantastic fuzzy lter of L. Using Theorem 58 we have
thatf1gis a fantastic lter of L, hence, by Theorem 64, Lis an MV-algebra.
34

Chapter 3
Fuzzy lter theory in residuated
lattices
3.1 Filters of residuated lattices
De nition 25. A residuated lattice is a structure L= (L;_;^; ;!;0;1)of type (2;2;2;2;2;0;0)
satisfying the following axioms:
(a27) (L;_;^;0;1)is a bounded lattice;
(a28) (L; ;1)is a commutative semigroup (with the unit element 1);
(a29) ( ;!;1)is an adjoint pair.
Aregular residuated lattice is a residuated lattice Lsatisfying the following regularity
equation, for all x2L:
(RL) (x!0)!0 =x.
In a resituated lattice L, for anyx2L,xnis inductively de ned as follows: x1=x,
xk+1=xk x,k2N,x=x!0,x= (x).
Proposition 9. In each residuated lattice L, the following assertions hold for all x;y;z2L:
(c17)x!(y!z) =y!(x!z);
(c18)x!y(y!z)!(x!z);
(c19)xy)yx;
(c20) 1!x=x,x!x= 1;
(c21) 0= 1,1= 0;
(c22)xy,x!y= 1;
(c23)x!(y^z) = (x!y)^(x!z),(x_y)!z= (x!z)^(y!z);
35

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
(c24)x!(y!(x y)) = 1 ;
(c25)x (y_z) = (x y)_(x z),(x_y)=x^y;
(c26)x yx^y,x x= 0;
(c27)yx!y,xx!y;
(c28)x_x= 1)x^x= 0;
(c29)x_y((x!y)!y)^((y!x)!x).
Proposition 10. In each regular residuated lattice L, the following properties hold for all
x;y2L:
(i)x!y=y!x;
(ii)x y= (x!y).
De nition 26. An MTL-algebra is a residuated lattice Lsatisfying the pre-linearity equa-
tion:
(PL) (x!y)_(y!x) = 1 , for allx;y2L.
An IMTL-algebra is an MTL-algebra Lsatisfying the regularity equation (RL).
A BL-algebra is an MTL-algebra Lsatisfying the following identity:
(BL)x^y=x (x!y), for allx;y2L.
An NM-algebra is an IMTL-algebra Lsatisfying the following identity:
(NM) (x y!0)_(x^y!x y) = 1 , for allx;y2L.
Lemma 7. Each BL-algebra is a distributive lattice.
As for MV-algebras , there are several equivalent de nitions. Particularly, it was proved
that an MV-algebra is equivalent to a BL-algebra Lsatisfying the following condition:
(MV) (x!y)!y= (y!x)!x, for allx;y2L.
There is another equivalent de nition of MV-algebras.
Lemma 8. LetLbe a residuated lattice. If Lsatis es the identity (MV), thenLis a regular
residuated lattice, and x_y= (x!y)!y, for allx;y2L.
Proof. For allx;y2L, by Proposition 9 and the condition ( MV) we have that 1 = y!
(x_y), and sox_y= 1!(x_y) = (y!(x_y))!(x_y) = ((x_y)!y)!y= ((x!
y)^(y!y))!y= (x!y)!y. This proves that x_y= (x!y)!y. Further, putting
y= 0, we obtain x=xfor allx2A, which means that Lis regular.
Lemma 9. LetLbe a residuated lattice. If Lsatis es the identity (MV), thenLis a
BL-algebra.
36

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Proof. Suppose that Lsatis es the identity ( MV), and letx;y2L. First we show that the
condition (BL) holds. Using Lemma 8 and Proposition 10, we obtain that x^y=x^y=
(x_y)= ((y!x)!x) = ((x!y)!x)=x (x!y). Next, we show that the
condition (PL) holds. Using ( PL) and (c17), we have that y!x= (y^(x_y))!x=
((y_x) ((x_y)!y))!x= ((x_y)!y)!((x_y)!x) = (x!y)!(y!x).
Hence it follows from Lemma 8 that ( x!y)_(y!x) = ((x!y)!(y!x))!(y!
x) = (y!x)!(y!x) = 1. Summing up the above results, we show that Lsatis es the
conditions ( BL) and (PL). Therefore Lis a BL-algebra.
Proposition 11. LetLbe a residuated lattice. Then Lis an MV-algebra if and only if L
satis es the condition (MV).
Proof. It is a consequence of Lemma 9.
De nition 27. A Boolean lattice, also called a Boolean algebra, is a bounded distributive
lattice (L;_;^;0;1)with a unary operation such that x_x= 1andx^x= 0for allx2L.
Proposition 12. LetLbe a residuated lattice. Then the following assertions are equivalent,
for allx;y2L:
(i) (x!y)!x=x;
(ii)x_x= 1;
(iii)x!x=x;
(iv)Lis a Boolean algebra.
Proof. (i))(ii):Suppose that ( i) holds and let x;y2L. Then, we have that ( y!x)!x=
(y!x)!((x!y)!x)(x!y)!y. By the symmetry, ( x!y)!y(y!x)!x.
This shows that Lsatis es the condition ( MV). Hence it follows from Lemma 8 that
x_x= (x!x)!x. Also, since x!x=x, by (i), we have that x_x=x!x= 1,
which means that ( ii) holds.
(ii))(iii):Suppose that ( ii) holds. Then we have that ( x_x)!x= 1!x. It follows
from Proposition 9 that x!x=x.
(iii))(i):Suppose that ( iii) holds. Since xx!y, we have that x!x(x!
y)!x. By (iii), this implies x(x!y)!x. Hence we have that ( x!y)!x=xby
(c27).
(i))(iv):Suppose that ( i) holds. Then the condition ( ii) also holds, and we see that
Lsatis es the condition ( MV) from the proof of (( i))(ii)). Thus, using Lemma 7 and 9,
we obtain that Lis a distributive lattice. Therefore it follows from ( ii) and (c28) thatLis
a Boolean algebra.
(iv))(ii):Is obvious.
De nition 28. LetXbe a set. Then, a function v:X![0;1]is called a fuzzy set on X.
Ifvis a fuzzy set on a set Xthen for 2[0;1], the setv :=fx2Xjv(x) gis called a
level subset (with level ) ofv.
37

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
LetXbe a set,Aa subset ofX. By the characteristic function of Ais meant the following
fuzzy setAonX:
A=(
1; x2A
0; x =2A:
Remark 14. For any fuzzy sets vandwon a setX, we write vwif and only if
v(x)w(x), for allx2X.
3.2 Filters of residuated lattices
In this section, we rst recall the de nition of lters and related results in general lattices.
Next we extend notions of the several particular lters naturally to general residuated lat-
tices, and further enumerate some relative results obtained in some particular classes of
residuated lattices, which still hold in general residuated lattices.
LetLbe a residuated lattice, FL,andx;y;z2L. For convenience, we enumerate
some conditions which will be used in the following study:
(c30)x;y2F)x y2F;
(c31)x2F;xy)y2F;
(c32) 12F;
(c33)x;x!y2F)y2F;
(c34)z;z!((x!y)!x)2F)x2F;
(c35)x_x2F;
(c36)x!(z!y);y!z2F)x!z2F;
(c37)z!(x!y);z!x2F)z!y2F;
(c38)x2!y2F)x!y2F;
(c39)x!x22F;
(c40)z;z!(y!x)2F)((x!y)!y)!x2F;
(c41)y!x2F)((x!y)!y)!x2F;
(c42) ((x!y)!y)!((y!x)!x)2F.
De nition 29. A nonempty subset FofLis called a lter of Lif it satis es the conditions
(c30)and(c31), for allx;y2L.
The set of all lters in Lis denoted by F(L).
Proposition 13. LetFbe a subset of L. ThenT2F(L)if and only if it satis es the
conditions (c32)and(c33), for allx;y2L.
38

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Theorem 68. LetFbe a subset of Land12F. Then the following assertions are equiva-
lent, for all x;y;z2L:
(i)Fis a lter of L;
(ii)x!y;y!z2F)x!z2F;
(iii)x!y;x z2F)y z2F;
(iv)x;y2F;xy!z)z2F.
LetFbe a lter of L. De ne relationFonLas follows:
xFy,x!y;y!x2F;8x;y2L:
It is easily veri ed that Fis a congruence relation. Let L=F denote the set of the
congruence classes of F, that means L=F :=f[x]Fjx2Lg, where [x]F:=fy2LjxFyg.
De ne
;*;u;tonL=F as follows:
[x]f
[y]f= [x y]F;[x]F*[y]F= [x!y]F
[x]Fu[y]F= [x^y]F;[x]Ft[y]F= [x_y]F:
Then (L=F;t;u;
;*;[0]F;[1]F) is a residuated lattice, which is called the quotient resid-
uated lattice with respect to F .
3.2.1 Boolean lters (implicative lters)
De nition 30. A subsetFofAis called an implicative lter of Aif it satis es the conditions
(c32)and(c34)for allx;y;z2A.
A subsetFofLis called a Boolean lter of Lif it is a lter of Lthat satis es the condition
(c35)for allx2L.
Proposition 14. LetFbe a subset of L. ThenFis an implicative lter of Lif and only if
Fis a Boolean lter of L.
Theorem 69. LetFbe a lter of L. Then the following assertions are equivalent, for all
x;y2L:
(i)Fis a Boolean lter of L;
(ii) (x!y)!x2F)x2F;
(iii) (x!x)!x2F;
(iv)The quotient residuated lattice L=F is a Boolean algebra.
Theorem 70. In any residuated lattice L, the following conditions are equivalent:
(i)Lis a Boolean algebra;
39

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
(ii)Any lter of Lis a Boolean lter of L;
(iii)f1gis a Boolean lter of L;
(iv) (x!y)!x=x, for allx;y2L.
Proof. By De nition 30, ( ii))(iii) is obvious, and ( i))(iii),(iv) is an immediate
consequence of Proposition 12.
3.2.2 G- lters (positive implicative lters)
De nition 31. A subsetFofLis called a positive implicative lter of Lif it satis es the
condition (c32)and(c37)for allx;y;z2L.
Remark 15. The positive implicative lter in De nition 31 is called an implicative lter.
De nition 32. A subsetFofLis called a G- lter of Lif it is a lter of Lthat satis es
the condition (c38)for allx;y2L.
Proposition 15. LetFbe a subset of L. ThenFis a positive implicative lter of Lif and
only ifFis a G- lter of L.
A BL-algebra Lis called a G odel algebra (brie
y, G-algebra), if x2=x x=xfor every
x2L. We now introduce the notion of a G(RL)-algebra as a generalization of G-algebras,
which is de ned as a residuated lattice Lsatisfyingx2=x x=xfor everyx2L.
Theorem 71. LetFbe a lter of L. Then the following assertions are equivalent, for all
x;y;z2L:
(i)Fis a G- lter of L;
(ii)z!(y!x)2F)(z!y)!(z!x)2F;
(iii)z;z!(y!(y!x))2F)y!x2F;
(iv)x!x22F;
(v)The quotient residuated lattice L=F is a G(RL)-algebra.
Theorem 72. In any residuated lattice L, the following assertions are equivalent:
(i)Lis G(RL)-algebra;
(ii)Any lter of Lis a G- lter;
(iii)f1gis a G- lter of L.
40

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
3.3 MV- lters
De nition 33. A subsetFofLis called a fantastic lter of Lif it satis es the conditions
(c32)and(c40)for allx;y;z2L. The fantastic lter will be called a normal (MP) lter.
De nition 34. A subsetFofLis called an MV- lter of Lif it is a lter of Lthat satis es
the condition (c41)for allx;y2L.
Proposition 16. LetFbe a subset of L. ThenFis a fantastic lter of Lif and only if F
is an MV- lter of L.
Theorem 73. LetFbe a lter of L. ThenFis an MV- lter of Lif and only if it satis es
the condition (c42)for allx;y2L.
Proof. Suppose that Fis an MV- lter of L, andx;y2L. By Proposition 9 we can easily
obtain that x!(x_y) = 12Fand (x_y)!y=x!y. Therefore it follows from ( c41)
that (((x_y)!y)!y)!(x_y)2F, that is, (( x!y)!y)!(x_y)2F. On the
other hand, since x_y(y!x)!x, we have that ( x_y)!((y!x)!x) = 12F.
Thus, by Theorem 68 it follows that (( x!y)!y)!((y!x)!x)2F, so the condition
(c42) is satis ed.
Conversely, suppose that Fsatis es the condition ( c42), and letx;y2Lsuch thaty!
x2F. Since (y!x)!(((x!y)!y)!x) = ((x!y)!y)!((y!x)!x)2F,
it follows from Proposition 13 that (( x!y)!y)!x2F. By De nition 34, this shows
thatFis an MV- lter of L.
Theorem 74. LetFbe a lter of L. ThenFis an MV- lter of Lif and only if the quotient
residuated lattice L=F is an MV-algebra.
Proof. It is obvious using Theorem 73.
Theorem 75. In any residuated lattice L, the following assertions are equivalent, for all
x;y2L:
(i)Lis an MV-algebra;
(ii)Any lter of Lis an MV- lter of L;
(iii)f1gis an MV- lter of L;
(iv) ((x!y)!y)!x=y!x;
(v) (x!y)!y= (y!x)!x.
3.4 Fuzzy lters of residuated lattices
LetAbe a residuated lattice, va fuzzy set on Landx;y;z2L. We enumerate some
assertions that will be further necessary:
(c43)v(x y)v(x)^v(y);
41

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
(c44)xy)v(x)v(y);
(c45)v(1)v(x);
(c46)v(y)v(x!y)^v(x);
(c47)v(x)(v(z!((x!y)!x)))^v(z);
(c48)v(x_x) =v(1);
(c49)v(x!z)(v(x!(z!y)))^v(y!z);
(c50)v(x!(x!y))v(x!y);
(c51)v(x!(x!y)) =v(x!y);
(c52)v(x!z)(v(x!(y!z)))^v(x!y);
(c53)v(x!x2) =v(1);
(c54)v(((x!y)!y)!x)v(y!x);
(c55)v(((x!y)!y)!x)(v(z!(y!x)))^v(z);
(c56)v(((x!y)!y)!((y!x)!x)) =v(1);
(c57)xy!z)v(z)v(x)^v(y).
De nition 35. A fuzzy set vofLis called a fuzzy lter of Lif it satis es the conditions
(c43)and(c44)for allx;y2L.
The set of all fuzzy lters of Lis denoted by FF(L).
Theorem 76. A fuzzy set vofLis a fuzzy lter of Lif and only if it satis es the conditions
(c45)and(c46)for allx;y2L.
Theorem 77. A fuzzy set vofLis a fuzzy lter of Lif and only if it satis es the condition
(c57).
Proposition 17. A fuzzy set vofLis a fuzzy lter of Lif and only if for any 2[0;1],
the level set v is either empty or a lter of L.
Corollary 4. LetFbe a nonempty subset of L. ThenFis a lter of Lif and only if the
characteristic function FofFis a fuzzy lter of L.
In addition, from Proposition 17, it follows that if vis a fuzzy lter of Lthenvv(1)is
always a lter of L.
Proposition 18. LetLbe a residuated lattice, va fuzzy lter of Landx;y2L. For any
z2Lwe de ne
vx:L![0;1];vx(z) =v(x!z)^v(z!x)
Thenvx=vyif and only if v(x!y) =v(y!x) =v(1):
42

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Proposition 19. LetLbe a residuated lattice, va fuzzy lter of LandL=v:=fvxjx2Lg
(the set of all cosets of v). For any vx;vy2L=v, we de ne
vxtvy=vx_y; vxuvy=vx^y; vx
vy=vx y; vx7!vy=vx!y:
ThenL=v= (L=v;t;u;
;7!;v0;v1)is a residuated lattice, or the fuzzy quotient residuated
lattice.
Theorem 78. Letvbe a fuzzy lter of L. then the residuated lattice L=v is isomorphic to
L=vv(1).
3.5 Fuzzy Boolean lters (fuzzy implicative lters)
De nition 36. A fuzzy lter vis called a fuzzy Boolean lter of L(brie
y, FB- lter), if it
satis es the condition (c48).
A fuzzy set vofLis called a fuzzy implicative lter of Lif it is a fuzzy lter of Lthat
satis es the condition (c49).
We will call the fuzzy implicative lter in De nition 36 a fuzzy positive implicative lter .
Proposition 20. Letvbe a fuzzy set on L. Thenvis an FB- lter of Lif and only if vis
a fuzzy implicative lter of L.
Theorem 79. Letvbe a fuzzy lter of L. Then the following assertions are equivalent, for
allx;y;z2L:
(i)vis an FB- lter of L;
(ii)v(x) =v(x!x);
(iii)v((x!y)!x)v(x);
(iv)v((x!y)!x) =v(x);
(v)v(x)(v(z!((x!y)!x)))^v(z);
(vi)v(x!z)v(x!(z!z));
(vii)v(x!z) =v(x!(z!z));
(viii)v(x!z)(v(y!(x!(z!z))))^v(y).
Corollary 5. A fuzzy set vofLis an FB- lter of Lif and only if it satis es the conditions
(c45)and(c47).
Proof. Using Theorem 79, the equivalence ( i),(v) is obvious. We only need to prove that
ifvsatis es the conditions ( c45) and (c47) thenvis a fuzzy lter of L. Puttingx=yin (c47),
we have that v(x)(v(z!((x!x)!x)))^v(z) = (v(z!(1!x)))^v(z) =v(z!
x)^v(z). Therefore vsatis es the condition ( c46). This and the condition ( c45) imply that
vis a fuzzy lter of L.
43

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Theorem 80. Letvbe a fuzzy of L. Then the following assertions are equivalent:
(i)vis an FB- lter of L;
(ii)L=vv(1)is a Boolean algebra;
(iii)L=v is Boolean algebra.
Theorem 81. In any residuated lattice L, the following assertions are equivalent, for all
x;y2L:
(i)Lis a Boolean algebra;
(ii)Every fuzzy lter of Lis an FB- lter of L;
(iii)f1gis an FB- lter of L.
Proof. (i))(ii) Suppose that Lis a Boolean algebra and let v2FF(L). Then we have
thatx_x= 1, for all x2A. Thusv(x_x) =v(1), which means that vis an FB- lter of
L.
(ii))(iii) It is obvious.
(iii))(i) Suppose that f1gis an FB- lter of L, for allx2L, then it follows that
f1g(x_x) =f1g(1). Since 12f1g, we obtain that f1g(x_x) = 1. This shows that
x_x2f1g, sox_x= 1. Hence, by Proposition 12, Lis a Boolean algebra.
3.6 Fuzzy G- lters (fuzzy positive implicative lters)
De nition 37. A fuzzy set vofLis called a fuzzy G- lter of L(brie
y, FG- lter), if it is
a fuzzy lter of Lthat satis es the condition (c50), for allx;y2L.
Remark 16. Obviously, the condition (c50)form De nition 37 could equivalently be replaced
with condition (c51).
De nition 38. Letvbe a fuzzy set on L. Thenvis called a fuzzy positive implicative lter
ofLif it satis es the conditions (c45)and(c52)for allx;y2L.
Proposition 21. Letvbe a fuzzy set of L. Thenvis an FG- lter of Lif and only if vis a
fuzzy positive implicative lter of L.
Theorem 82. Letvbe a fuzzy lter of L. Then the following assertions are equivalent for
allx;y;z2L:
(i)vis an FG- lter of L;
(ii)v(x!(y!z))v((x!y)!(x!z));
(iii)v(x!(y!z)) =v((x!y)!(x!z));
(iv)v((x y)!z) =v((x^y)!z).
44

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Theorem 83. Letvbe a fuzzy lter of L. Thenvis an FG- lter of Lif and only if v
satis es the condition (c53)for allx2A.
Proof. Suppose that vsatis es the condition ( c63), and letx;y2A. Since
(x!x2)!(x!y)x2!y=x!(x!y);
it follows from ( c44) thatv((x!x2)!(x!y))v(x!(x!y)). Moreover, by ( c43),
(c46) and (c53) we have that
v(x!y)(v((x!x2)!(x!y)))^(v(x!x2))
(v(x!(x!y)))^(v(x!x2))
= (v(x!(x!y)))^(v(1))
=v(x!(x!y)):
This proves that vsatis es the condition ( c50). Therefore vis an FG- lter of Lby
De nition 37.
Conversely, suppose that vis an FG- lter of L.Sincex!(x!(x x)) = 1, by ( c24), we
have thatv(x!(x!(x x))) =v(1). Hence it follows from ( c52) thatv(x!(x x)) =
v(1), which means that v(x!x2) =v(1).
3.7 Fuzzy MV- lters (fuzzy fantastic lters)
De nition 39. A fuzzy set vofLis called a fuzzy MV- lter of L(brie
y, FMV- lter), if
it is a fuzzy lter of Lthat satis es (c54)for allx;y2L.
Remark 17. In De nition 39 the condition (c54)could equivalently be replaced with the
following condition :
(c58)v(((x!y)!y)!x) =v(y!x), for allx;y2L.
Theorem 84. A fuzzy set vofLis an FMV- lter of Lif and only if it satis es the conditions
(c45)and(c55).
Proof. Suppose that vis an FMV- lter of L. Then, it follows that vsatis es the conditions
(c45), (c46) and (c54). Moreover, by ( c46) and (c54) we have
v(((x!y)!y)!x)v(y!x)(v(z!(y!x)))^(v(z)):
This shows that vsatis es the condition ( c55).
Conversely, suppose that vsatis es the conditions ( c45) and (c55). On the one hand,
takingz= 1 in (c55), we obtain that vsatis es the condition ( c54). On the other hand,
takingy= 1 in (c55), we have that
v(x) =v(((x!1)!1)!x)
(v(z!(1!x)))^(v(z))
= (v(z!x))^(v(z));
for allx;z2L. This shows that vsatis es the condition ( c46). Sincevalso satis es the
condition (c45), it follows from Theorem 76 that vis a fuzzy lter of L.
45

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Theorem 85. Letvbe a fuzzy lter of L. Thenvis an FMV- lter of Lif and only if v
satis es the condition (c56)for allx;y2L.
Proof. Suppose that vsatis es the condition ( c56) for allx;y2L. Since
((x!y)!y)!((y!x)!x) = (y!x)!(((x!y)!y)!x);
it follows form ( c45) and (c46) that
v(((x!y)!y)!x)(v(y!x))^(v(((x!y)!y)!((y!x)!x)))
= (v(y!x))^(v(1))
=v(y!x):
This proves vsatis es the condition ( c54). Therefore vis an FMV- lter.
Conversely, suppose that vis an FMV- lter of L. We rst prove that vv(1)is an MV- lter
ofL. In fact, let x;y2Lbe such that y!x2vv(1), thenv(y!x)v(1). This and ( c54)
imply that v(((x!y)!y)!x)v(1), which means that (( x!y)!y)!x2vv(1).
By De nition 34, this shows that vv(1)is an MV- lter. Therefore it follows from Theorem
73 that ((x!y)!y)!((y!x)!x)2vv(1), that means v(((x!y)!y)!((y!
x)!x))v(1). This and ( c45) imply that vsatis es the condition ( c56).
3.8 Regular lters
De nition 40. A ltervofLis called a regular lter of L(brie
y, R- lter), if it satis es
the following condition:
(RF)x!x2Ffor allx2L.
The following example shows that R- lters exists in a residuated lattice.
Example 5. LetL:= [0;1]:For anya;b2Lwe de ne:
a_b=maxfa;bg; a^b=minfa;bg;
a!b=(
1; ifab
(1a)_b;otherwise:
a b=(
0; ifa+b1
a^b;otherwise:
Then we can easily check that (L;^;_; ;!;0;1)is a residuated lattice, and F:= (c;1]is
an R- lter of L, where 0:5c1.
Theorem 86. LetFbe a lter of L. Then the following assertions are equivalent:
(i)Fis an R- lter of L;
(ii)x!y2F)y!x2Ffor allx;y2L;
46

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
(iii)x!y2F)y!x2Ffor allx:y2L.
Proof. (i))(ii) Suppose that Fis an R- lter of L. ThenFis a lter of Land satis es the
condition (RF). For allx;y2L, using Proposition 9, we have that x!yy!x
y!x, and so it follows that ( x!y)!(y!x)(y!x)!(y!x)x!x.
This implies that the following equality holds ( x!x)!((x!y)!(y!x)) = 1.
Since 12Fandx!x2F, by Proposition 13 we have that the condition ( c69) holds.
(ii))(i) Suppose that Fsatis es the condition ( ii) and letx2L. Sincex=x, we
have thatx!(x)=x!x= 12F. Thus, we obtain that x!x2Fby the
condition (ii). This shows that Fis an R- lter of L.
(i))(iii)Suppose that Fis an R- lter of Land letx;y2A. Using Proposition 9, we
have thatx!yy!x, and so the following inequality holds:
(x!y)!(y!x)(y!x)!(y!x)x!x:
Thus, according to the methods used in the above demonstration, we can prove that the
condition (iii) holds.
(iii))(i) Suppose that Fsatis es the condition ( iii). Sincex!x= 12Ffor all
x2L, it follows that x!x2F. HenceFis an R- lter of L.
Corollary 6. In a residuated lattice L, the lter generated by the subset fx!xjx2Ag
is the least R- lter of L.
Corollary 7. LetFbe an R- lter of L, thenx2Fif and only if x2F, for allx2L.
Corollary 8. (Extension property) If Fis an R- lter of L, then every lter Gcontaining
Fis also an R- lter of L.
Remark 18. For Boolean lters, G- lters and MV- lters, the corresponding extension prop-
erties also hold.
Theorem 87. LetFbe a subset of L. ThenFis an R- lter if and only if it satis es the
following conditions, for all x;y;z2L:
(i) 12F;
(ii)z;z!(x!y)2F)y!x2F.
Proof. Suppose that Fis an R- lter of L. ThenFis a lter of L, and the condition ( i) holds.
Now we show that Fsatis es (ii). For any x;y;z2L, letz2Fndz!(x!y2F.
Then, it follows from ( c33) thatx!y2F, and so we have that y!x2Fby Theorem
86. Consequently, this proves that the condition ( ii) holds.
Conversely, suppose that Fsatis es the condition ( i) and (ii). Now we prove that Fis
an R- lter of L. By Theorem 86 and Proposition 13, it suces to show that Fsatis es the
condition (ii) from Theorem 86 and ( c33). First, putting z= 1 in (ii), we obtain that F
satis es (c49) by (i). Next, let x;y2Lbe such that x!y2Fandx2F. Sincex!y
y!xx!yy!x=y!x, we have that x!yy!x=x!y.
Therefore, it follows that
(x!y)!((x)!(y)) = (x!y)!(x!y) = 1:
47

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Thus, we have that x!y2Fimpliesy!x2Fby the condition ( i) and (ii).
Particularly, 1!x=x2Fimpliesx=x!12F. Summing up the above
results, we obtain that x!y2Fandx2Fimplyx!y2Fandx2F. Since
x!y=x!(y!1), it follows from ( ii) thaty= 1!y2F. This proves that F
satis es (c43).
Theorem 88. LetFbe a subset of L. Then,Fis an R- lter if and only if it satis es the
following conditions, for all x;y;z2L:
(i) 12F;
(ii)z;z!(x!y)2F)y!x2F.
Proof. It is similar to the proof of Theorem 87.
Proposition 22. LetFbe a lter of L. Then the quotient residuated lattice L=F is regular
if and only if Fis an R- lter of L.
Proof. The proof is straightforward using the de nition of regular residuated lattices and
R- lters.
3.9 Fuzzy regular lters
De nition 41. A fuzzy lter vofLis called a fuzzy regular lter of L(brie
y, FR- lter),
if it satis es the following condition:
(FR)v(x!x) =v(1)for allx2L.
Corollary 9. (Extension property). Let vandwbe fuzzy lters of Lwithvwand
v(1) =w(1). Ifvis an FR- lter of L, thenwis also a lter of L.
Proof. The proof is straightforward from De nition 41.
Remark 19. For FB- lters, FG- lters and FMV- lters, the corresponding extension prop-
erties are hold too.
Theorem 89. Letvbe a fuzzy lter of L. Then the following conditions are equivalent:
(i)vis an FR- lter of L;
(ii)v(x!y)v(y!x)for allx;y2L;
(iii)v(x!y)v(y!x)for allx;y2L.
Proof. (i))(ii) Suppose that vis an FR- lter of L.Thenvis a fuzzy lter of L, and satis es
the condition ( FR). For allx;y2L, using Proposition 9, we have that x!yy!
xy!x, and so the following inequality holds for all x;y2L:
(x!y)!(y!x)(y!x)!(y!x)x!x:
48

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Thus, by (c44) and Theorem 76 we have that v(y!x)(v((x!y)!(y!x)))^
(v(x!y))(v(x!x))^(v(x!y)) = (v(1))^(v(x!y)) =v(x!y), and so
(ii) holds.
(ii))(i) Suppose that vsatis es the condition ( ii) and letx2L. Sincex=x, we
have thatx!(x)=x!x= 1, and so v(x!(x)) =v(1). Thus, by condition
(ii) we obtain that v(x!x)v(1), and so v(x!x) =v(1) by (c45). This shows that
vis an FR- lter of L.
(i))(iii) Suppose that vis an FR- lter of Land letx;y2L. Using Proposition 9, we
have thatx!yy!x, and so we have the following inequality for all x;y2L:
(x!y)!(y!x)(y!x)!(y!x)x!x:
Thus, according to the methods of proving ( i))(ii), we can obtain that v(x!y)
v(y!x) for allx;y2L.
(iii))(i) Suppose that vsatis es the condition ( iii). Sincex!x= 1 for allx2L, it
follows from ( iii) thatv(x!x)v(x!x) =v(1), and so v(x!x) =v(1) by (c55).
Hencevis an FR- lter of L.
Remark 20. In Theorem 89 the conditions (ii)and(iii)could equivalently be replaced by
the following conditions:
(i)v(x!y) =v(x!y)for allx;y2L;
(ii)v(x!y) =v(y!x)for allx;y2L.
Corollary 10. Ifvis an FR- lter of L, thenv(x) =v(x)for allx2L.
Proof. For allx2L, sincevis an FR- lter of L, it follows from Remark 20 ( ii) that
v(x) =v(x!0) =v(0!x) =v(1!x) =v(x):
Theorem 90. A fuzzy set vofLis an FR- lter of Lif and only if it satis es
(i)v(1)v(x)for allx2L;
(ii)v(y!x)(v(z!(x!y)))^(v(z))for allx;y;z2L.
Proof. Suppose that vis an FR- lter of L. Thenvis a fuzzy lter of L, and the condition
(i) holds. Now we only need to prove that vsatis es (ii). In fact, by Theorem 76 we see
that for any x;y;z2L,
v(x!y)(v(z!(x!y)))^(v(z));
and by Theorem 89 we have that v(y!x)v(x!y). Consequently, we deduce that
v(y!x)(v(z!(x!y)))^(v(z)), thusvsatis es (ii).
Conversely, suppose that a fuzzy set vsatis es the conditions ( i) and (ii). Takingz= 1
in (ii), we have that vsatis es (ii) from Theorem 89 by ( i). Now, we show that vsatis es
(c46). Letx;y2A. Sincex!y=x!(1!y) =x!(0!y), it follows from ( ii)
that (v(x!y))^(v(x)) = (v(x!(0!y)))^(v(x))v(y!0) =v(y). Similarly,
sincey= 1!y= 1!y= 1!(y!0), it follows from ( c45) and (ii) that
49

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
v(y) = (v(1!(y!0)))^(v(1))v(0!y) =v(1!y) =v(y). Summing up the
above results, we have that v(y)(v(x!y))^(v(x)), which means that vsatis es the
condition (c46). Hence,vis a fuzzy lter of Lby Theorem 76, and also it is an FR- lter of
Lby Theorem 89.
Theorem 91. A fuzzy set vofLis an FR- lter of Lif and only if it satis es the following
condition :
(i)v(1)v(x)for allx2L;
(ii)v(y!x)(v(z!(x!y)))^(v(z))for allx;y;x2L.
Proof. It is similar to the proof of Theorem 90.
Theorem 92. A fuzzy set vofLis an FR- lter of Lif and only if for any 2[0;1],v is
either empty or an R- lter of L.
Proof. Suppose that vis an FR- lter of Landv 6=;for some 2[0;1]. Then, there
is an element x02v such thatv(x0) . Sincevis a fuzzy lter of L, we have that
v(1)v(x0) by (c45), and sov(1) . This implies that vv(1)v . In addition, it follows
from Proposition 17 that both vv(1)andv are lters of L. Now we prove that vv(1)is an R-
lter.Letx!y2vv(1)for allx;y2L. Thenv(x!y) =v(1). Thenv(x!y) =v(1).
Sincevis an FR- lter, we have that v(x!y) =v(y!x) by Remark 20( ii). Hence we
have thatv(y!x) =v(1), and so y!x2vv(1). This shows that vv(1)is an R- lter of
Lby Theorem 86. Using the extension properties of R- lters, we obtain that v is also an
R- lter.
Conversely, suppose that for each 2[0;1],v is either empty or an R- lter of L.
Since 12vv(1)(6=;), it follows that vv(1)is certainly an R- lter of L. Thus, we have that
x!x2vv(1)for allx2L, and sov(x!x) =v(1). This shows that vis an FR- lter
ofL.
Theorem 93. LetFbe an R- lter of L. Then there exists an FR- lter vofLsuch that
v =Ffor some 2(0;1).
Proof. Letvbe a fuzzy set on Lde ned by
v(x) :=(
;x2F
0; x =2F;
where is a xed number 0 < < 1. Now we verify that vis an FR- lter of L. For this
purpose, let x;y;x2Lbe such that z!(x!y)2Fandz2F. Theny!x2Fby
Proposition 13. Consequently, we deduce that
v(z!(x!y)) =v(z) =v(y!x) = ;
and sov(y!x) = (v(z!(x!y)))^(v(z)). If at least one of z!(x!y) and
zis not inF, then at least one of v(z!(x!y)) andv(z) is 0. Hence, we obtain
thatv(y!x)(v(z!(x!y)))^(v(z)). Summing up the above results, we have
v(y!x)(v(z!(x!y)))^(v(z)), for allx;y;z2L. On the other hand, since 1 2F,
we havev(1) = v(x), for allx2L. By Theorem 90, this shows that vis an FR- lter of
L. Clearly,v =F.
50

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Corollary 11. LetFbe an nonempty subset of L. ThenFis an R- lter of Lif and only if
the characteristic function Fis an FR- lter of L.
Corollary 12. Letvbe a fuzzy lter of L. Thenvis an FR- lter of Lif and only if the
level setvv(1)is an R- lter of L.
Remark 21. Replacing the R- lter/FR- lter by the Boolean lter/FB- lter (respectively,
G- lter/FG- lter, MV- lter/FMV- lter), the corresponding Theorems 92, 93, Corollaries
11 and 12are all true, and their proofs are also similar, respectively.
3.10 Characterizations of some particular algebras
Theorem 94. In residuated lattice L, the following assertions are equivalent :
(i)Lis a regular residuated lattice;
(ii)Filterf1gis an R- lter of L;
(iii)Every lter of Lia an R- lter;
(iv)x!y=y!xfor allx;y2L;
(v)x!y=y!xfor allx;y2L;
(vi)The quotient L=F is a regular residuated lattice for every F2F(L).
Proof. (i))(ii) Suppose that Lis a regular residuated lattice. Then, for all x2Lwe have
thatx=x, and sox!x2f1g. Sincef1gis a lter of L, it follows thatf1gis an
R- lter ofL.
(ii))(iii) It follows immediately using Corollary 8.
(iii))(i) Suppose that every lter of Lis an R- lter. Specially, we have that f1gis an
R- lter ofL. Hencex!x2f1gfor allx2Lthat isx!x= 1, which means xx.
This imply x=xfor allx2L. Therefore Lis a regular residuated lattice.
(i))(iv) Suppose that Lis a regular residuated lattice and let x;y2L. Then, we have
thatx!yy!x=y!x. This and ( c20) implyx!y=y!x.
(iv))(i) Suppose that the condition ( iv) holds. Then, by putting y= 1 in (iv), we
obtain that x=x!0 =x!1= 1!x=x. This shows that Lis a regular residuated
lattice.
(i))(v) Suppose that the condition ( i) holds. Then the condition ( iv) also holds. Hence,
it follows that x!y=y!x=y!x, for allx;y2L, this means the condition ( v)
holds.
(v))(i) The proof is similar to that ( iv))(i).
(i))(vi) It follows directly from Proposition 22.
Theorem 95. In a residuated lattice L, the following assertions are equivalent:
(i)Lis a regular residuated lattice;
(ii)Every fuzzy lter of Lis an FR- lter;
51

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
(iii)Fuzzy lter f1gis an FR- lter;
(iv)vv(1)is an R- lter of Lfor allv2FF(L);
(v)v(x!y) =v(y!x)for allx;y2Land allv2FF(L);
(vi)v(x!y) =v(y!x)for allx;y2Land allv2FF(L).
Proof. It follows from Theorem 89 and Remark 20 that ( ii),(v),(vi). (ii),(iv) is
obtained from Corollary 12, and it is obvious that ( i))(ii))(iii). For the implication
(iii))(i) suppose that the condition ( iii) holds, that means f1gis an FR- lter of L.
Hence, we have that f1gis an R- lter of Lby Corollary 11, and so Lis a regular residuated
lattice by Theorem 94.
Theorem 96. Letvbe a fuzzy lter of L. Then the fuzzy quotient L=v is a regular residuated
lattice if and only if vis an FR- lter of L.
Proof. Suppose that vis an FR- lter of L, Thenv(x!x) =v(1) for allx2L. And also,
xximplies that v(x!x) =v(1) for allx2L. Using Proposition 18, we have that
vx=vx, that means ( vx!v0)!v0=vx. This implies that the fuzzy quotient L=vis a
regular residuated lattice.
Conversely, suppose that the fuzzy quotient L=v is a regular residuated lattice. Then
(vx!v0)!v0=vxfor allx2L. Using Proposition 18 again, we obtain that v(x!
x) =v(1), which means that vis an FR- lter of L.
Corollary 13. A residuated lattice Lis regular if and only if the fuzzy quotient L=v is
regular for every v2FF(L).
Theorem 97. Letvbe a fuzzy lter of L. Thenvis an FG- lter of Lif and only if the
fuzzy quotient residuated lattice L=v is a G(RL)-algebra.
Proof. By Theorem 71 and Remark 21, we have that vis an FG- lter of Lif and only if
vv(1)is a G- lter of Lif and only if L=vv(1)is a G(RL)-algebra. By Theorem 78, this is also
equivalent to L=vis a G(RL)-algebra.
Theorem 98. In any residuated lattice L, the following assertions are equivalent, for all
x;y2L:
(i)Lis a G(RL)-algebra;
(ii)Every fuzzy lter of Lis an FG- lter of L;
(iii)f1gis an FG- lter of L;
(iv)vv(1)is an G- lter of Lfor allv2FF(L);
Proof. The proof is an immediate consequence of Theorem 72 and Remark 21.
Theorem 99. Letvbe a fuzzy lter of L. Thenvis an FMV- lter of Lif and only if the
fuzzy quotient residuated lattice L=v is an MV-algebra.
52

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Proof. The proof is similar to that of Theorem 97.
Theorem 100. In any residuated lattice L, the following assertions are equivalent, for all
x;y2L:
(i)Lis an MV-algebra;
(ii)Every fuzzy lter of Lis an FMV- lter of L;
(iii)f1gis an FMV- lter of L.;
(iv)vv(1)is an MV- lter of Lfor allv2FF(L);
Proof. It is an immediate consequence of Theorem 75 and Remark 21.
Proposition 23. A residuated lattice Lis Boolean algebra if and only if vv(1)is a Boolean
lter ofLfor allv2FF(L).
3.11 The relations among special lters
(Fuzzy) Boolean lters (MV- lters, R- lters, G- lters) are some important types of (fuzzy)
lters in residuated lattices. We have established the relation between these lters and their
corresponding special fuzzy lters in Remark 21. However, we can similarly prove that the
same results also hold in general residuated lattices.
Theorem 101. In an residuated lattice L, a lterFofLis a Boolean lter if and only if
it is both G- lter and MV- lter.
Theorem 102. In an residuated lattice L, a fuzzy lter vofLis a fuzzy Boolean lter if
and only if it is both FG- lter and FMV- lter.
Remark 22. In a residuated lattice, (fuzzy) Boolean lters are certainly (fuzzy) G- lters and
(fuzzy) MV- lters, but a (fuzzy) G- lter of (fuzzy) MV- lter may not be a (fuzzy) Boolean
lter.
Theorem 103. In residuated lattice L, every FMV- lter is an FR- lter, but the converse
does not hold.
Proof. Suppose that vis an FMV- lter of Land letx2L. Thevis a fuzzy lter of Lthat
satis es the condition ( c54). Since 0!x= 1 andx!x= ((x!0)!0)!x, it follows
from (c54) that
v(x!x) =v(((x!0)!0)!x)v(0!x) =v(1):
This and (c45) meanv(x!x) =v(1). So,vis an FR- lter of L.
In the following example we will prove that not every FR- lter is an FMV- lter.
53

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Example 6. In Example 5, for the residuated lattice (L;^;_; ;!;0;1)is an R- lter of
L, where 0:5c1. However, letting 0:8c1, thenFis not an MV- lter because
0:3!0:8 = 12F, but ((0:8!0:3)!0:3)!0:8 = 0:8=2F. Thus, it follows from
Corollary 11 that Fis an FR- lter, but not an FMV- lter.
Theorem 104. In an BL-algebra L, FMV- lter and FR- lters are equivalent.
Proof. By Theorem 103, we need only to prove that any FR- lter is an FMV- lter in an
BL-algebra L. Suppose that vis an FR- lter of L, andx;y2L. Then, it follows from
Remark 20 that vis a fuzzy lter and v(y!x) =v(x!y). Sincex!y=x!
(x^y) =x!(y (y!x)) by the condition ( BL), we have that
v(y!x) =v(x!(y (y!x)))
=v(x!(y (x!y)))
=v((y (x!y))!x)
=v(((y (x!y))!0)!0)
=v(((x!y)!(y!0))!x)
=v(((x!y)!y)!x):
And, since
(((x!y)!y)!x)!(((x!y)!y)!x)
((x!y)!y)!((x!y)!y)
(x!y)!(x!y)
y!y;
we have that
v((((x!y)!y)!x)!(((x!y)!y)!x))v(y!y) =v(y!y) =v(1):
In addition, since (( x!y)!y)!x((x!y)!y)!x, we have that
v(((x!y)!y)!x)v(((x!y)!y)!x):
Thus, it follows from ( c46) that
v(((x!y)!y)!x)v(((x!y)!y)!x)
(v((((x!y)!y)!x)!(((x!y)!y)!x)))^(v(((x!y)!y)!x))
(v(1))^(v(y!x))
=v(y!x):
Summing up the above results, we obtain that v(y!x)v(((x!y)!y)!x) for all
x;y2L. By De nition 39, this proves that vis an FMV- lter of L.
Theorem 105. LetLbe a residuated lattice, va fuzzy set of L. Thenvis a fuzzy Boolean
lter ofLif and only if it is both an FG- lter and an FR- lter.
Proof. It follows from Theorem 102 and 103 that every FB- lter is both an FG- lter and an
FR- lter.
54

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Conversely, suppose that vbe both an FG- lter and an FR- lter of L. Now we prove that
vis an FB- lter of L. For this purpose, according to Theorem 79, we need only to show that
vsatis es the following condition:
v(x)v((x!y)!x);for allx;y2L:
First, for any x;y2L, sincexx!y, we have that ( x!y)!xx!x. By (c54),
this means that v((x!y)!x)v(x!x). Next, since x!x=x!x= (x)!x,
it follows from Remark 16 and Corollary 10 that v(x!x) =v(x!0) =v(x) =v(x).
Summing up the above results, we obtain that v(x)v((x!y)!x) for allx;y2L.
From Theorems 105 and 95, we have the following results:
Corollary 14. In regular residuated lattices, FB- lters and FG- lters are equivalent.
Since IMTL-algebras, R0-algebras, MV-algebras and lattice implication algebras are all
particular cases of regular residuated lattices, by Corollary 14 we obtain the following:
Corollary 15. FB- lters and FG- lters are equivalent in IMTL-algebras, R0-algebras, MV-
algebras or lattice implication algebras.
Corollary 16. FB- lters, FMV- lters and FR- lters are equivalent in G(RL)-algebras.
Proof. It is an immediate consequence of Theorems 98, 102 and 105.
Now we describe some relations among R- lters, MV- lters, G- lters and Boolean lters.
This results are all similar to the cases of fuzzy lters, and they can be obtained from the
above corresponding results about fuzzy lters, respectively. This suciently show that
fuzzy lters are useful tool to obtain results on classical lters.
Theorem 106. In a residuated lattice L, every MV- lter is an R- lter, but the converse
does not hold.
Proof. Suppose that Fis an MV- lter of L. Then, by Remark 21 we obtain the Fis an
FMV- lter of L. Moreover, it follows from Theorem 103 that Fis an FR- lter, and so F
is an R- lter by Corollary 11. This proves that every MV- lter is an R- lter in a residuated
lattice. On the other hand, Example 6 shows that the converse does not hold.
Theorem 107. In BL-algebras, MV- lters and R- lters are equivalent.
Proof. By Theorem 106, we need only the proof that any R- lter is an MV- lter in a BL-
algebraL. Suppose that Fis an R- lter of L. Then, it follows from Corollary 11 that F
is an FR- lter. Moreover, it follows from Theorem 104 that Fis an FMV- lter. Hence, by
Remark 21 we obtain that Fis an MV- lter of L.
Using the similar methods in the proofs of Theorem 106 and 107 we can easily obtain
from the following results on classical lters.
Theorem 108. LetLbe a residuated lattice, Fa subset of L. ThenFis a Boolean lter if
and only if it is both a G- lter and an R- lter.
55

CHAPTER 3. FUZZY FILTER THEORY IN RESIDUATED LATTICES
Corollary 17. In regular residuated lattice, Boolean lters and G- lters are equivalent.
Corollary 18. Boolean lters and G- lters are equivalent in IMTL-algebras, R0-algebras,
MV-algebras or lattice implication algebras.
Corollary 19. Boolean lters, MV- lters and R- lters are equivalent in G(RL)-algebras.
Finally, using the relations among Boolean lters, MV- lters and R- lters, we can obtain
the following relations among the corresponding special residuated lattices.
Theorem 109. LetLbe a residuated lattice. Then Lis a Boolean algebra if and only if it
is a regular G(RL)-algebra.
Proof. It is an immediate consequence of Theorems 70, 72, 94 and Corollary 17.
56

Chapter 4
Positive implicative pseudo-valuations
on BCK-algebras
4.1 Preliminaries
De nition 42. An algebra (A;!;1)of type (2;0)is called a BCI-algebra if it satis es the
following assertions:
(i) (x!y)!((y!z)!(x!z)) = 1 ;
(ii)x!((x!y)!y) = 1 ;
(iii)x!x= 1;
(iv)8x;y2A,x!y=y!x= 1)x=y.
If a BCI-algebra Asatis es the following identity:
(v)8x2A,x!1 = 1 ,
thenAis called a BCK-algebra.
Proposition 24. Any BCK-algebra Asatis es the following axioms:
(i)8x;y;z2A,xy)z!xz!y;y!zx!z;
(ii)8x;y;z2A,x!(y!z) =y!(x!z);
(iii)8x;y;z2A,(z!x)!(z!y)x!y.
We can de ne a partial ordering byxyif and only if x!y= 1.
De nition 43. A nonempty subset Dof a BCK-algebra Ais called an ideal of Aif it satis es
the following assertions, for all x;y2A:
(i) 02D;
(ii)x;y!x2D)y2D:
A nonempty subset Dof a BCK-algebra Ais called a positive implicative ideal of Aif it
satis es (i)from De nition 43 and
(iii)x!(y!z)2D;x!y2D)x!z2D;8x;y;z2A.
57

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
4.2 Positive implicative pseudo-valuations on BCK-algebras
De nition 44. ([9]) A real-valued function 'onAis called a weak pseudo-valuation on A
if it satis es the following condition, for all x;y2A:
'(y)'(x)'(x!y): (4.1)
De nition 45. ([9]) A real-valued function 'onAis called a pseudo-valuation on Aif it
satis es the following conditions, for all x;y2A:
'(1) = 0; (4.2)
'(y)'(x)'(x!y): (4.3)
A pseudo-valuation 'onAsatisfying the condition
8x2A;x6= 1)'(x)6= 0; (4.4)
is called a valuation on A.
Proposition 25. For any pseudo-valuation 'onA, we have the following inequalities:
(i)'(x)0for allx2A;
(ii)'is order preserving;
(iii)'(x!y)'(x!z) +'(z!y)for allx;y;z2A.
Proposition 26. Every pseudo-valuation 'onAsatis es the following implication:
8x;y;z2A;x!(y!z) = 1)'(x)'(y) +'(z): (4.5)
Proof. Letx;y;z2Abe such that x!(y!z) = 1. It follows from (4.2) and (4 :3) that
'(x!y)'(x!(y!z)) +'(z) ='(1) +'(z) ='(z)
so that'(x)'(x!y) +'(y)'(y) +'(z).
Corollary 20. Let'be a pseudo-valuation on A. If
a1!(:::!(an1!(an!x)):::) = 1;
then'(x)nX
k=1'(ak).
Theorem 110. Every real-valued function 'onAsatisfying conditions (4:2)and(4:5)is a
pseudo-valuation on A.
Proof. Sincex!((x!y)!y) = 1 for all x;y2A, it follows from (4 :5) that'(y)
'(x!y) +'(x). Therefore 'is a pseudo-valuation on A.
Proposition 27. For a real-valued function 'onA, has the following properties:
58

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
(i)'(x!y)'(x!(x!y));
(ii)'((z!x)!(z!y))'(z!(x!y));
(iii)'(x!y)'(z!(x!(x!y))) +'(z).
Proposition 28. If'is a pseudo-valuation on A, then (i)and(ii)from Proposition 27 are
equivalent.
Proof. (i))(ii) Assume that ( i) is true. Then
x!(x!((x!y)!z)) =x!((x!y)!(x!z))x!(y!z)
and using ( i) from Proposition 24 and ( i) be obtain
'((x!y)!(y!z)) ='(x!((x!y)!z))'(x!(x!((x!y)!z)))'(x!(y!z)):
(ii))(i) Is obtained replacing zwithyin (ii) and using ( iii) from De nition 42 and ( i)
from Proposition 24.
De nition 46. A real-valued function 'onAis called a positive implicative pseudo-valuation
onAif it satis es (4:2)and
'(x!z)'(x!(y!z)) +'(x!z);8x;y;z2A: (4.6)
De nition 47. A positive implicative pseudo-valuation 'onAsatisfying the condition 4.4
is called a positive implicative valuation on A.
A mapping ':A!R;x7!1is a positive implicative valuation on A, called trivial
positive implicative pseudo-valuation on A.
Example 7. LetA=f1;a;bgbe a BCK-algebra with the following operation table:
! 1 a b
11 a b
a1 1 b
b1 1 1.
Let'be a real-valued function on Ade ned by
'=1a b
1 5 7
:
Then'is a positive implicative pseudo-valuation on A.
Example 8. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c
11 a b c
a1 1 b a
b1 a 1 c
c1 1 b 1.
59

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Let'be a real-valued function on Ade ned by
'=1a b c
1 1 1 1
:
Then'is a positive implicative pseudo-valuation and a pseudo valuation on A.
Theorem 111. Every positive implicative pseudo-valuation on Ais a pseudo-valuation on
D.
Proof. Let':D!Rbe a positive implicative pseudo-valuation on A. Takingz= 1
in (4:6) and using ( i) from Proposition 24, then '(y)'(x!y) +'(x), hence'is a
pseudo-valuation on A.
We will now show with some examples that the converse of Theorem 111 may not be true.
Example 9. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c
11 a b c
a1 1 b a
b1 a 1 c
c1 1 b 1
Let'be a real-valued function on Ade ned by
'=1a b c
1 3 1 2
:
Then'is a p pseudo-valuation on A, but not a positive implicative pseudo-valuation on
A, because'(a!c) ='(a) = 3'(a!(a!c)) +'(a!a) ='(a!a) +'(1) =
'(1) +'(1) = 2 .
Example 10. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c
11 a b c
a1 1 a c
b1 1 1 c
c1 a b 1
Let'be a real-valued function on Ade ned by
'=1a b c
1 4 4 7
:
Then'is a p pseudo-valuation on A, but not a positive implicative pseudo-valuation on
A, because'(a!b) ='(a) = 4'(a!(a!b)) +'(a!a) ='(a!a) +'(1) =
'(1) +'(1) = 2 .
60

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
For any real-valued function 'onA, we consider the following set:
I':=fx2Aj'(x) = 0g:
Theorem 112. If a real-valued function 'onAis a (positive implicative) pseudo-valuation
onA, then the set I'is a (positive implicative) ideal of A.
Proof. Suppose'is a pseudo-valuation on A. Then, 02I'. Letx;y2Abe such that
y2I'andy!x2I'. Then'(y) = 0 and '(y!x) = 0. It follows from (4 :3) that
'(x)'(y!x) +'(y) = 0, so using Proposition 25 ( i) we obtain that '(x) = 0. Hence
x2I', and soI'is an ideal of A.
Suppose'is a positive implicative pseudo-valuation on A. Then 02I'. Letx;y;z2Abe
such thatx!(y!z)2I'andx!y2I'. Then'(x!(y!z)) = 0 and '(x!y) = 0.
Using equation (4 :6), we have
'(x!z)'(x!(y!z)) +'(x!y) = 0;
and so, by Proposition 25 ( i),'(x!z) = 0, which means x!z2I', therefore I'is a
positive implicative ideal of A.
We will now show, by some examples, that the converse of Theorem 112 may not be true.
So we will prove that there exists a BCK-algebra Aand a mapping ':A!Rsuch that
(i)'is not a (positive implicative) pseudo-valuation on A,
(ii)I':=fx2Aj'(x) = 0gis a (positive implicative) ideal of A.
Example 11. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c d
11 a b c d
a1 1 b c c
b1 a 1 c d
c1 1 1 1 a
d1 1 1 1 1
Let'be a real-valued function on Ade ned by
'=1a b c d
1 4 1 7 2
:
ThenI'=f1;bgis an ideal of A, but'is not a pseudo-valuation on Abecause'(c) =
7'(c!d) +'(d) ='(a) +'(d) = 4 + 2 = 6 .
Example 12. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c d
11 a b c d
a1 1 b a d
b1 a c 1 d
c1 1 b 1 d
d1 a 1 c 1
61

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Let#be a real-valued function on Ade ned by
#=1a b c d
1 1 8 1 3
:
ThenI'=f1;a;cgis a positive implicative ideal of A, but#is not a positive implicative
pseudo-valuation on Abecause#(c!b) =#(b) = 8#(c!(b!d))) +#(c!d) =#(c!
d) +#(d) =#(d) + 2 = 2 + 2 = 4 .
We will now use two examples to show that a pseudo-valuation on Adoes not necessary
have to satisfy conditions of Proposition 27.
Example 13. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c
11 a b c
a1 1 b a
b1 a 1 c
c1 1 b 1
Let'be a real-valued function on Ade ned by
'=1a b c
1 3 1 2
:
'does not satisfy Proposition 27 (i)because'(a!c) ='(a) = 3'(a!(a!c)) =
'(a!a) ='(1) = 1 .
'does not satisfy Proposition 27 (ii)because'((a!a)!(a!c)) ='(1!a) = 3
'(a!(a!c)) ='(a!a) ='(1) = 1 .
'does not satisfy Proposition 27 (iii)because'(a!c) ='(a) = 3'(1!(a!(a!
c)))) +'(1) ='(1!(a!a)) +'(1) ='(1!1) +'(1) ='(1) +'(1) = 2 .
Example 14. LetA=f1;a;b;cgbe a BCK-algebra with the following operation table:
! 1 a b c
11 a b c
a1 1 a c
b1 1 1 c
c1 a b 1
Let'be a real-valued function on Ade ned by
'=1a b c
1 4 4 7
:
It is obvious that '(a!b) ='(a) = 4'(a!(a!b)) ='(a!a) ='(1) = 1 and
'((a!a)!(a!b)) ='(1!a) ='(a) = 4'(a!(a!b)) ='(a!a) ='(1) = 1 ,
and so'does not satisfy the conditions (i)and(ii)from Proposition 27.
Since'(a!b) ='(a) = 4'(1!(a!(a!b))) +'(1) ='(1!(a!a)) ='(1!
1) ='(1) = 1 , then'does not satisfy the condition (iii)Proposition 27 either.
62

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Proposition 29. Every positive implicative pseudo-valuation 'onAsatis es the conditions
from Proposition 27.
Proof. Let'be a positive implicative pseudo-valuation on A. Then, by Theorem 111, 'is
a pseudo-valuation on A. Takingz=yin equation (4 :6), we obtain
'(x!y)'(x!(x!y)) +'(y!y)
='(x!(x!y)) +'(1)
='(x!(x!y))
for allx;y2A. This shows that 'satis es the condition ( i) Proposition 27. By Proposition
28,'also satis es Proposition 27 ( ii).
Now, using De nition 42 ( iii), Proposition 24 ( i), (iii), equation (4 :3) and Proposition 27
(ii) we obtain
'(x!y)'(x!(y!z)) +'(z)
='((y!y)!(y!(x!z))) +'(z)
'(y!(y!(z!x))) +'(z)
='(x!(y!(y!z))) +'(z)
for allx;y;z2A.
Theorem 113. If a pseudo-valuation 'onAsatis es either the condition (i)or(ii)from
Proposition 27, then 'is a positive implicative pseudo-valuation on A.
Proof. Let'be a positive-valuation on Awhich satis es the condition ( i) Proposition 27.
Then, for all x;y;x2A, (x!z)!(x!(x!z))y!(x!z) =x!(y!z). Since
'is order preserving, then
'((x!y)!(x!(z!x)))'(x!(y!z)):
Using Proposition 27 ( i) and equation (4 :3) we have
'(x!z)'(x!(x!z))
'((x!y)!(x!(x!z))) +'(x!y)
'(x!(y!z)) +'(x!y):
so we conclude that 'is positive implicative pseudo-valuation on A.
Let'be a pseudo-valuation on Awhich satis es Proposition 27 ( ii). Then, using (4 :3)
and Proposition 27 ( ii) for allx;y;z2A, we have
'(x!z)'((x!y)!(x!z)) +'(x!y)
'(x!(y!z)) +'(x!y):;
therefore'is a positive implicative pseudo-valuation on A.
Theorem 114. Let'be a real-valued function on A. If'satis es (4:3)and Proposition
27(iii), then'is a positive implicative pseudo-valuation on A.
63

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Proof. Assume that 'satis es (4:2) and Proposition 27 ( iii), then for all x;z2A, we have
'(x) ='(x!1)'(x!(z!(z!1)))) +'(z) ='(x!z) +'(z) which means 'is a
pseudo-valuation on A.
Now we take z= 1 in Proposition 27 ( iii). Using Proposition 24 ( i) and (4:2) for all
x;y2A, we obtain
'(x!y)'(x!(y!(y!1))) +'(1) ='(x!(x!y)):
Furthermore, by Theorem 113, it follows that 'is a positive implicative pseudo-valuation
onA.
Proposition 30. Every positive implicative pseudo-valuation 'onAsatis es the condition
a1!(a2!(x!(x!y))) = 1)'(x!y)'(a1) +'(a2); (4.7)
for allx;y;a 1;a22A.
Proof. By Theorem 111 we know that 'is a pseudo-valuation. Assume that a1!(a2!
(x!(x!y))) = 1 for all x;y;a 1;a22A, then using Proposition 26, we have '(x!(x!
y))'(a1) +'(a2). It follows from De nition 42 ( iii), Proposition 24 ( i) andc72) that
'(x!y) ='(x!(y!1))
='((x!1)!(x!y))
'(x!(x!y))
'(a1) +'(a2):
Therefore the proof is complete.
Theorem 115. Every real-valued function 'onAthat satis es the conditions (4:2)and
(4:7)is a positive implicative pseudo-valuation.
Proof. Letx;y;z2Abe such that x!(y!z) = 1. Then x!(y!(z!(z!1))) = 1.
Using (4:7) it follows that '(x) ='(x!1)'(y) +'(z), thus, by Theorem 109, 'is a
pseudo-valuation on A.
We only need to prove that 'is a positive implicative pseudo-valuation on A. For this,
sincex!((y!(y!1))!(y!(1!y))) = 1 for all x;y2A, it follows from (4 :2) and
(4:7) that
'(x!y)'(x!(x!y)) +'(1) ='(x!(x!y)):
Furthermore, using Theorem 113, we obtain that 'is a positive implicative pseudo-valuation
onA.
Now we o er a generalization of Proposition 30.
Proposition 31. Every positive implicative pseudo-valuation 'onAsatis es the condition
a1!(:::!(an!(x!(x!y))):::) = 1)'(x!y)nX
i=1'(ai): (4.8)
for allx;y;a 1;a2;:::;an2A.
64

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Proof.'is a pseudo-valuation on Aby Theorem 111, it follows using Proposition 29 and
Corollary 20 that
'(x!y)'(x!(x!y))nX
k=1'(ak)
which completes the proof.
Proposition 32. Every positive implicative pseudo-valuation 'onAsatis es the equation
a!(b!(x!(y!z))) = 1)'((x!y)!(x!z))'(a) +'(b); (4.9)
for allx;y;x;a;b2A.
Proof. Assume that a!(b!(x!(y!z))) = 1 for all x;y;z;a;b2A, then, using
Propositions 26 and 29, we obtain
'((x!y)!(x!z))'(x!(y!z))'(a) +'(b)
which completes the proof.
Theorem 116. If a real-valued function 'onAsatis es equation (4:2)and(4:9)then'is
a positive implicative pseudo-valuation on A.
Proof. Assume that a!(b!(x!(y!z))) = 1 for all x;y;z;a;b2A, then, using
Proposition 24 ( i), (4:9) and De nition 42 ( iii), we obtain
'(x!y) ='(x!(y!1)) ='((x!y)!(y!y))'(a) +'(b):
Now, by Theorem 115 it follows that 'is a positive implicative pseudo-valuation on A.
Using the above results, we give some characterizations of positive implicative pseudo-
valuation.
Theorem 117. If'is a real-valued function on A, then the following assertions are equiv-
alent:
(i)'is a positive implicative pseudo-valuation on A;
(ii)'is a pseudo-valuation on Athat satis es Proposition 27 (i);
(iii)'is a pseudo-valuation on Athat satis es Proposition 27 (ii);
(iv)'satis es conditions (4:2)and Proposition 27 (iii);
(v)'satis es conditions (4:2)and(4:7);
(vi)'satis es conditions (4:2)and(4:9).
Theorem 118. Let'and be pseudo-valuation on Asuch that'(x) (x)for allx2A.
If is a positive implicative pseudo-valuation on A, then so is '.
65

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Proof. Using Proposition 24 ( i), Proposition (29) and De nition 42 ( iii), we obtain
'((x!(y!z))!((z!y)!(z!z)))
='((x!y)!((x!(y!z))!(x!z)))
='((x!y)!(x!((x!(y!z))!z)))
 ((x!y)!(x!((x!(y!z))!z)))
 (x!(x!((x!(y!z))!z)))
= (x!((x!(y!z))!(y!z)))
= ((x!(y!z))!(x!(y!z)))
= (1) = 1:
Now, using equation (4 :3) we have
'((z!y)!(z!x)'((z!(y!x))!((z!y)!(z!x))) +'(z!(y!x))
1 +'(z!(y!x)) ='(z!(y!x))
so, by Theorem 113 it follows that 'is a positive implicative pseudo-valuation on A.
4.3 Fuzzy lters in BCI-algebras
De nition 48. An ideal of a BCI-algebra is a subset Icontaining 0such that if y!x2I
andy2I, thenx2I.
If the algebra is commutative, then an ideal Iis prime if it proper and if whenever x^y2I,
thenx2Iory2I.
An idealIis maximal if it is proper and whenever IJfor some ideal J, thenI=Jor
Jis the whole algebra.
De nition 49. A nonempty set Fof a BCI-algebra Xis said to be a lter if
(i)x2Fandxy)y2F;
(ii)x2Fandy2F)x^y2Fandy^x2F.
A lterFis prime it is proper and if whenever x_y2F, thenx2Fofy2F.
A lterFis maximal if it is proper and whenever TUfor some lter U, thenF=U
orUis the whole algebra.
De nition 50. A fuzzy set of a BCI-algebra Ais a function :X7![0;1].
is a fuzzy ideal if it satis es (0)(x)and(x)((xy);(x)).
A fuzzy ideal is prime if (x^y) =max((x);(y)).
We can de ne a partial ordering relation on a set of all fuzzy ideal of Abyif
and only if (x)(x)for allx2A.
A fuzzy ideal is maximal if it is a maximal element of the set of all fuzzy ideals of A.
De nition 51. [4?] A fuzzy set of a commutative BCI-algebra Ais a fuzzy lter if it
satis es:
(x^y)min((x);(y))
and whenyx, we have
(y)(x);8x2xandy2A:
66

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Proposition 33. A fuzzy subset of a commutative BCI-algebra Ais a fuzzy lter if and
only if(x^y) =(x)^(y), for allx;y2A.
Example 15. Every constant function :A7![0;1]is a fuzzy lter.
Example 16. LetA=f0;1;2;3gwith!de ned by the following table:
! 0 1 2 3
00 1 2 3
10 0 1 2
20 0 0 1
30 0 0 0
It is easy to check that Ais a commutative BCI-algebra. Let be a fuzzy subset on X
de ned by(1) =(3) =(2)> (0) =(1). By routine calculations we can prove that 
is a fuzzy lter of A..
Example 17. LetA=f0;a;b;c; 1gwith!de ned by the following table:
! 0 a b c 1
00 a b c 1
a0 a a c c
b0 0 0 c c
c0 0 a 0 a
10 0 0 0 0
Letbe a fuzzy subset on Ade ned by(a=(b) =(o)<(c) =(1). Now it is easy to
prove thatis a fuzzy lter of A.
Given a fuzzy subset andt2[0;1], we de ne t=fx2Xj(x)tg.is a fuzzy
lter if and only if tis either empty or a lter. Thus, given a fuzzy lter of A, the set
A=fx2Xj(x) =(1)gis a lter.
IfFis a lter, then the characteristic function of F,Fis a fuzzy lter. Moreover, given
a lterFofA,
AF=fx2AjF(x) = 1g=F:
4.4 Fuzzy prime lter in MV-algebra
In this section, by Awe will denote a bounded commutative BCK-algebra.
De nition 52. A fuzzy lter of an MV-algebra Ais prime if it is non constant and
(x_y) =(x)_(y);8x;y2A:
Fis a lter of Aif and only if the characteristic function of Fis a fuzzy lter.
Theorem 119. A lterFof an MV-algebra Ais prime if and only if the characteristic
function of F,Fis a fuzzy prime lter.
67

CHAPTER 4. POSITIVE IMPLICATIVE PSEUDO-VALUATIONS ON
BCK-ALGEBRAS
Theorem 120. A fuzzy lter of an MV-algebra Ais prime if and only if the set
t=fx2Aj(x)tg
is either empty or a prime lter of A.
Proof. Suppose that is a fuzzy lter of A, thentis a fuzzy lter. Let x_y2t. Then
(x^y)t. Sinceis a prime lter, then (x
veey) =(x)_(y). So(x)tor(y)t. We have that x2tory2(t), which proves
thattis prime.
Conversely, suppose that t=fx2Aj(x)tgis a prime lter of A, thentis a fuzzy
lter. Letx;y2Aandt=(x_y);x_y2t. Sincetis a prime lter, we have x2tor
y2(t). So(x)tor(y)tand we obtain
(x)_(y)t=(x_y):
On the other hand, xx_yandyx_y. Now we apply the De nition 51 and
obtain that (x)(x_y) and(y)(x_y) so(x)_(x)(x_y). Finally
(x)_(x) =(x_y) and we consider that is a fuzzy prime lter.
De nition 53. [4?] Letbe a fuzzy subset of Aand 2[0;1]. The function
 :A7![0;1]
is given by  = ((x)) :
68

Bibliography
[1] R. Balbes, P. Dwinger, Distributive Lattices , Univ. of Missouri Press, Columbia, Missouri, 1974.
[2] K. Blount, C. Tsinakis, The structure of residuated lattices , Intern. J. Alg. Comp, 13(2003), 437-461.
[3] C. Bu sneag, Valuation on residuated lattices , An. Univ. Craiova Ser. Mat. Inform., 34(2007), 21-28.
[4] D. Bu sneag, Categories of Algebraic Logic , Editura Academiei Rom^ ane, Bucharest, 2007.
[5] D. Bu sneag, A note on deductive systems of a Hilbert algebra , Kobe J.Math, 2(1985), 29-35.
[6] D. Bu sneag, Hilbert algebras of fractions and maximal Hilbert algebras of quotients , Kobe J. Math,
5(1988), 161-172.
[7] D. Bu sneag, Hertz algebras of fractions and maximal Hertz algebras of quotients , Kobe J. Math,
39(1993), 461-469.
[8] D. Bu sneag, On extensions of pseudo-valuations of Hilbert algebras , Discrete Math, 263(2003), 11-24.
[9] D. Bu sneag, Hilbert algebras with valuations , Math. Jpn., 44(1996), 285-289.
[10] S. Celani, A note on homomorphism of Hilbert algebras , Int. J. Math. Sci., 29(2002), 55-61.
[11] I. Chajda, R. Hala s,Congruences and ideals in Hilbert algebras , Kyungpook Math. J., 39(1999), 429-
432.
[12] I. Chajda, R. Hala s,Stabilizers in Hilbert algebras , Multiple-valued logic in Eastern Europe, Multi-
Valued Log., 8(2002), 139-148.
[13] I. Chajda, R.Hala s, Y.B. Jun, Annihilators and deductive systems in commutative Hilbert Algebras ,
Comment. Math. Univ. Carolin., 43(2002), 404-417.
[14] C.C. Chang, Algebraic analysis of many valued logics , Trans. Am. Math. Soc., 88(1958), 467-490.
[15] R. Cignoli, I.M.L. D'Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning , Kluwer,
Dordrecht, 2000.
[16] P.S. Das, Fuzzy groups and level subgroups , J.Math. Anal. Appl., 84(1991) 264-269.
[17] A. Diego, Sur les algebras de Hilbert, in Hermann , Ed.Collection de Logique Math, Serie A, Vol. 21
(1966).
[18] M.I. Doh, M.S. Kang, BCK/BCI-algebras with pseudo-valuation , Honam Math. J., 32(2010), 217-226.
[19] W.A. Dudek, Y.B. Jun, On fuzzy ideals in Hilbert algebras , Novi Sad J. Math., 29(1999), 193-207.
[20] W.A. Dudek, Y.B. Jun, Remarks on fuzzy topological subalgebras in Hilbert algebras , Filomat, 13(1999),
15-23.
[21] A. Dvure censkij, J. Rach _ unek, Probabilistic averaging in bounded commutative residuated l-monoids ,
Discrete Mathematics. 306(2006), 1317-1326.
[22] G. Dymek, Fuzzy prime ideals of pseudo-MV-algebras , Soft Comput., 12(2008), 365-372.
[23] N. Galatos, P.Jipsen, T.Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural
Logics , Elsevier Studies in Logic and Foundations, 2007.
69

BIBLIOGRAPHY
[24] B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, The pseudo-linear semnatics of interval-valued
fuzzy logics , Inform. Sci., 179(2009), 717-728.
[25] P. Hajek,Metamathematics of Fuzzy Logic , Kluwer, Dordrecht, 1998.
[26] P. Hajek,Basic fuzzy logic and BL-algebras ,Soft Comput., 2(1998), 124-128.
[27] M. Haveshki, A.B. Saeid, E. Eslami, Some types of Filters in BL-algebras , Soft Comput., 10(2006),
657-664.
[28] U. Hohle,Commutative, residuated l-monoids , Kluwer, Dordrecht, 1995, 53-106.
[29] C.S. Hoo, Fuzzy ideals of BCI and MV-algebras , Fuzzy Sets and Syst., 62(1994), 111-114.
[30] C.S. Hoo, Fuzzy implicative and Boolean ideals of MV-algebras , Fuzzy Sets and Syst., 66(1994), 315-327.
[31] P.M. Idziak, Lattice operations in BCK-algebras , Mathematica Japonica, 29(1984), 839-846.
[32] P. Jipsen, C. Tsinakis, A survey of residuated lattices. Ordered algebraic structures , Kluwer, Dordrecht,
2002, PP. 19-56.
[33] Y.B. Jun, Commutative Hilbert algebras , Soochow J. Math., 22(1996), 477-484.
[34] Y.B. Jun, Remarks on fuzzy topological subalgebras in Hilbert algebras , Filomat, 13(1999), 15-23.
[35] Y.B. Jun, S.M. Hong Fuzzy deductive systems of Hilbert algebras ,Indian J. pure appl. Math., 27(1996),
141-151.
[36] Y.B. Jun, Extensions of fuzzy deductive systems in Hilbert algebras , Fuzzy Sets and Systems, 79 (1996),
263-265.
[37] Y.B. Jun, Deductive systems of Hilbert algebras , Math. Japon, 43(1996), 51-54.
[38] Y.B. Jun, Characterizations of Noetherian BCK-algebras via fuzzy ideals , Fuzzy Sets and Systems,
108(1999), 231-234.
[39] Y.B. Jun, S.M. Hong, E.H. Roh, BCI-semigroups , Honam Math. J., 15(1993), 59-64.
[40] Y.B. Jun, K.H. Kim, Intuitionistic fuzzy ideals of BCK-algebras , Int. J. Math. Sci, 24(2000), 839-849.
[41] Y.B. Jun, E.H. Koh, Fuzzy p and -ideals in IS-algebras , J. Fuzzy Math., 7(1999), 473-480.
[42] Y.B. Jun, A. Valendziak, Fuzzy ideals of pseudo-MV-algebras , Inter. rev. Fuzzy Math., 1(2006), 21-31.
[43] Y.B. Jun, Fuzzy positive implicative and fuzzy associative lters of lattice implication algebras , Fuzzy
Sets and Syst., 121(2001), 353-357.
[44] Y.B. Jun, S.Z. Song, On fuzzy implicative lters of lattice implication algebras , J. Fuzzy Math., 10(2002),
893-900.
[45] Y.B. Jun, S.Z. Song, Fuzzy n-fold positive implicative lters in lattice implication algebras , J. Appl.
Math. Comput., 13(2003), 153-163.
[46] Y.B. Jun, S.Z. Song, On fuzzy fantastic lters of lattice implication algebras , J. Appl. Math. Comput.,
14(2004), 137-155.
[47] Y.B. Jun, Y.Xu, X.H.Zhang, Folding theory of implicative/fantastic lters in lattice implication algebras ,
Commun. Korean Math. Soc., 19(2004), 11-21.
[48] Y.B. Jun, Y. Xu, X.H. Zhang, Fuzzy lters of MTL-algebras , Inform.Sci. , 175(2005), 120-138.
[49] Y.B. Jun, Weak fuzzy implication algebras , Adv. Stud. Contemp. Math. 7(2003), no.1, 41-52.
[50] Y.B. Jun, Weak and concrete lters of WFI-algebras , J. Appl. Math. Informatics 26(2008), 925-932.
[51] K.H. Kim, Q. Zhang, Y.B. Jun, On fuzzy lters of MTL-algebras , J.Fuzzy Math., 10(2002), 981-989.
[52] M. Kondo, W.A. Dudek, Filter theory of BL-algebras , Soft Comput., 12(2008), 419-423.
70

BIBLIOGRAPHY
[53] M. Kondo, W.A. Dudek, On transfer principle in fuzzy theory , Mathware and Soft. Comput., 13(2005),
41-55.
[54] J.J. Lai, Y. Xu, J. Ma, On extension of LI-ideal in lattice implication algebras , Theor. Adv. Appl. Fuzzy
Logic ASC, vol.42, Springer-Verlag, Berlin, Heidelberg, 2007, 337-348.
[55] L.Z. Liu, K.T. Li, Fuzzy implicative and Boolean lters of R0-algebras , Inform. Sci., 171(2005), 61-67.
[56] L.Z. Liu, K.T. Li, Fuzzy lters of BL-algebras , Inform. Sci., 173(2005), 141-154.
[57] L.Z. Liu, K.T. Li, Fuzzy Boolean and positive implicative lters of BL-algebras , Fuzzy Sets and Syst.,
152(2005), 333-348.
[58] Y.L. Liu, M.Y. Ren, Normal MP- lters of R0-algebras , Fuzzy Information and Engineering ASC, vol.
54, Springer-Verlag, Berlin, Heidelberg, 2009, 113-118.
[59] X.L. Ma, J.M. Zhan, Y. Xu, Generalized fuzzy lters of R0-algebras , Soft. Comput., 11(2007), 1079-1087.
[60] X.L. Ma, J.M. Zhan, Y. Xu, Generalized fuzzy lters of BL-algebras , Appl. Math. J. Chin. Univ., Ser.
B, 22(2007), 490-496.
[61] X.L. Ma, J.M. Zhan, On(2;2^q)-fuzzy lters of BL-algebras , Jrl. Syst. Complexity, 21(2008), 144-158.
[62] X.L. Ma, J.M. Zhan, Y.B. Jun, On(2;2^q)-fuzzy lters of R0-algebras , Math. Log. Quart., 55(2009),
452-467.
[63] X.L. Ma, J.M. Zhan, W.A.Dudek, , Some kinds of (2;2^q)-fuzzy lters of BL-algebras , Comput Math,
Appl, 58(2009), 248-256.
[64] D.S. Malik and J.N. Mordeson, Extensions of fuzzy subrings and fuzzy ideals , Fuzzy Sets and Systems,
45(1992), 245-251.
[65] J. Meng, Y.B. Jun, Bck-algebras , Kyungmoon Com., Seoul, 1994.
[66] N.P. Mukherjee, Fuzzy Groups:Some group theoretic analogs , Inform.Sci., 39(1986), 247-268.
[67] D.W. Pei, The characterization of residuated lattices and regular residuated lattices , Acta Math. Sci.,
Ser. A, 45(2002), 271-278.
[68] D.W. Pei, G.J. Wang, The completeness and aplication of the formal system L, Sci. China Ser.F,
45(2002), 40-50.
[69] D.W. Pei, On equivalent forms of fuzzy logic systems NM and IMTL , Fuzzy Sets and Syst., 138(2003),
187-195.
[70] D. Piciu, Algebras of Fuzzy Logic , Editura Universitaria, Craiova, 2007.
[71] J. Rach _ unek, A duality between algebras of basic logic and bounded representable DRI-monoids , Math.
Bohem., 126(2001), 561-569.
[72] J. Rach _ unek, D. Salounova,Boolean deductive systems of bounded commutative residuated l-monoids ,
Contrib. Gen. Algebra, 16(2005), 199-208.
[73] J. Rach _ unek, D. Salounova,local bounded commutative residuated l-monoids , Czech. Math. J., 57(2007),
395-406.
[74] J. Rach _ unek, D. Salounova,Classes of lters in generalization of commutative fuzzy structures , Acta
Univ.Palacki. Olomuc. Fac. Rer. Nat., Math. to appear.
[75] J. Rach _ unek, D. Salounova,Fuzzy lters and fuzzy prime lters of bounded Rl-monoids and pseudo
BL-algebras , Inform. Sci., 178(2008), 3474-3481.
[76] J. Rach _ unek, D. Salounova, Classes of fuzzy lters of resituated lattice ordered monoids , Mathematica
Bohemica, No.1, 81-97
[77] J. Rach _ unek, V. Slez ak,Negation in bounded commutative DRl-monoids , Czech. Math. J., 56(2006),
755-763.
71

BIBLIOGRAPHY
[78] E.H. Roh, Y.B. Jun, W. H. Shim, Fuzzy associative -ideals of IS-algebras , Int. J. Math., Math Sci.,
24(2000), 729-735.
[79] A. Rosenfeld, Fuzzy groups , J.Math. Anal. Appl. 35(1971), 512-517.
[80] J.G. Shen, X.H. Zhang, Filters of residuated lattices , Chin. Quart. J. Math., 21(2006), 443-447.
[81] E. Turunen, Mathematics Behind Fuzzy Logic , Physica-Verlag, Heidelberg, 1999.
[82] E. Turunen, Boolean deductive systems of BL-algebras , Arch. Math. Logic, 40(2001), 467-473.
[83] G.J. Wang, MV-algebras, BL-algebras, R0-algebras and multiple valued logic , Fuzzy Syst. Math.,
16(2002), 1-15.
[84] M. Ward, R.P. Dilworth, Residuated lattices , Trans. Amer. Math. Soc., 45(1939), 335-354.
[85] Y. Xu, K.Y. Qin, On lters of lattice implication algebras , J. Fuzzy Math., 1(1993), 251-260.
[86] Y. Xu, D. Ruan, K. Qin, J. Liu, Lattice-Valued Logic , Springer-Verlag, Berlin, Heidelberg, 2003.
[87] L.A. Zadeh, Fuzzy sets , Inform. and Control, 8(1965), 338-353.
[88] L.A. Zadeh, The concept of a lingvistic variable and its application to approximate reason , Inform.
Control, 18(1975), 199-249.
[89] J.M. Zhan, Y.B. Jun, Notes on rede ned fuzzy implicative lters of lattice implication algebras , Inform.
Sci., 179(2009), 3182-3186.
[90] J.M. Zhan, Y. Xu, Some types of generalized fuzzy lters of BL-algebras , Comput. Math. Appl.,
56(2008), 1604-1616.
[91] J. Zhan, W.A. Dudek, Y.B. Jun, Interval valued (2;2^q-fuzzy lters of pseudo BL-algebras , Soft.
Comput., 13(2009), 13-21.
[92] L.A. Zhan, Z. Tan, Intuitionistic fuzzy deductive systems in Hilbert algebras , Southeast Asian Bulletin
of Math., 29(2005), 813-826.
[93] L.A. Zhan, Z. Tan, T-fuzzy multiply positive implicative BCC-ideals of BCC-algebras , Int. J. Math. ,
Math. Sci., 42(2003), 2653-2666.
[94] X.H. Zhang, W.H. Li, On fuzzy logic algebraic system MTL , Adv. Systs. Sci. Appl., 5(2005), 475-483.
[95] X.H. Zhang, Y.Q. Wang, Y.L. Liu, Fuzzy ultra lters and fuzzy G- lters of MTL-algebras , Lecture Notes
in Computer Science, LNAI, vol. 3613, Springer-Verlag, Berlin, Heidelberg, 2005, 160-166.
[96] X.M. Zhang, Y.B. Jun, M.I. Doh, On fuzzy lters and fuzzy ideals of BL-algbebras , Fuzzy Syst. Math.,
3(2006), 8-20.
[97] X.H. Zhang, On lters in MTL-algebras , Adv. Syst. Sci. Appl. , 7(2007), 32-38.
[98] Y.Q. Zhu, Q. Zhang, E.H. Roh, On fuzzy implicative lters of implicative systems , Southeast Asian
Bull. Math., 27(2003), 761-767.
72

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: 2 BCK-algebras and fuzzy theories 4 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . [611998] (ID: 611998)

Dacă considerați că acest conținut vă încalcă drepturile de autor, vă rugăm să depuneți o cerere pe pagina noastră Copyright Takedown.

Similar Posts