Filter theory in comm utativ e and non-comm utativ e residuated [611997]

Filter theory in comm utativ e and non-comm utativ e residuated
lattices
Phd Dana Mihaela Bolta³u
Decem b er 10, 2019
Abstract
The aim of this pap er is to dev elop a lattice theory and sho w ho w it c hange if the residuated
lattice is comm utativ e or non-comm utativ e. Moreo v er, w e mak e some c haracterization of lters
using lattice prop erties.
Keywor ds : residuated lattice, lter, implicativ e lter, p ositiv e implicativ e, b o olean, fan tastic
1 In tro duction
In researc h of logics, theory of lters pla ys a v ery imp ortan t role in pro ving completeness with
resp ect to algebraic seman tics.Residuation is a fundamen tal concept of ordered structures and cat-
egories. They ha v e b een in v estigated b y Krull (1924), Dilw orth (1939),W ard and Dilw orth (1939),
W ard (1940), Balb es and Dwinger (1974) and P a v elk a (1979). In Idziak (1984) pro v es that the
class of residuated lattices is equational. These lattices ha v e b een kno wn under man y names: BCK-
lattices in Idziak (1984), full BCK-algebras in Ok ada and T erui (1999), F Lew-algebras in Ono
and K omori (1985), and in tegral, residuated, comm utativ e monoids in H o hle (1995). W ard (1940),
W ard and Dilw orth (1939) w ere the rst who in tro duced the concept of a residuated lattice as a
generalization of ideal lattices of a ring. In their original denition, a residuated lattice is what
w e w ould call it an in tegral comm utativ e one. The general denition of a residuated lattice w as
giv en b y Galatos et al.(2007). They rst dev elop ed the structural theory of this kind of algebra
ab out residuated lattices. Ov er the last ten y ears, with the computers and information in science
dev eloping rapidly , the residuated lattice theory made great progress. Man y exp erts and sc holars
had carried out thorough systematical researc h in to it, they studied it from dieren t p oin ts of view.
F or example, Bloun t and T sinakis (2003) to ok the residuated lattice theory as an expansion of the
theory of groups; J. S. Olson (2008) in v estigated it from the view of v ariet y; Galatos et al. (2007)
in v estigated it from the view of semiring. The theory of residuated lattices w as used to dev elop
algebraic coun terparts of fuzzy logics in T urunen (1999) and substructural logics in H. Ono (2003).
Ha jek (1998) in v estigated the notion of BL-algebras and the concepts of lters and prime lters
in BL-algebras in order to pro vide an algebraic pro of of the completeness theorem of Basic Logic
(BL, for short), arising from the con tin uous triangular norms, familiar in the fuzzy logic frame-
w ork. Using prime lters in BL-algebras, he pro v ed the completeness of Basic Logic. So on after,
T urunen (1999) published a study on BL-algebras and their deductiv e systems. A w eak er logic
than BL called Monoidal T-norm Based Logic (MTL, for short) w as dened b y Estev a and Go do
(2001) and pro v ed b y Jenei and Mon tagna (2002) to b e the logic of left con tin uous T-norms and
their residua. The algebraic coun terpart of this logic is MTL-algebra, also in tro duced b y Estev a
and Go do (2001). In Estev a and Go do (2001) a residuated lattice L is called MTL-algebra if the
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prelinearit y prop ert y holds in L. Recen tly , T urunen and Mertanen (2008) and D. Bu³neag (2013)
dened the notion of semi-divisible residuated lattice and in v estigated their prop erties. Also, D.
Bu³neag (2015) in v estigated the notion of Stonean residuated lattices and they discussed it from
the view of ideal theory .
Residuated lattices are v ery useful and are basic algebraic structures. Man y logical algebras,
suc h as Bo olean algebras, MV-algebras, BL-algebras, MTL-algebras, G o del algebras, NM- algebras,
andR0 -algebras, are particular residuated lattices. Besides their logical in terest, residuated lattices
ha v e lots of in teresting prop erties. Filters pla y a vital role in in v estigating logical algebras and the
completeness of the corresp onding nonclassical logics.
2 Preliminaries
Denition 2.1. ([1],[2],[5],[7],[8],[18]) A c ommutative r esiduate d lattic e is a structur e L= (L;_;^;;!
;0;1) of typ e (2;2;2;2;2;0;0) satisfying the fol lowing axioms:
(CRL 1) (L;_;^;0;1) is a b ounde d lattic e;
(CRL 2) (L;;1) is a c ommutative semigr oup (with the unit element 1);
(CRL 3) (;!;1) is an adjoint p air.
A r e gular r esiduate d lattic e is a residuated lattice L satisfying the follo wing regularit y equation,
for allx2L :
(RL) (x!0)!0 =x .
Prop osition 2.1. ([5],[6],[7],[8],[19] In e ach c ommutative r esiduate d lattic e L , the fol lowing asser-
tions hold for al l x;y;z2L :
(c1)x!(y!z) =y!(x!z) ;
(c2)x!y(y!z)!(x!z) ;
(c3)xy)yx;
(c4) 1!x=x ,x!x= 1 ;
(c5) 0= 1 ,1= 0 ;
(c6)xy,x!y= 1 ;
(c7)x!(y^z) = (x!y)^(x!z) ,(x_y)!z= (x!z)^(y!z) ;
(c8)x!(y!(xy)) = 1 ;
(c9)x(y_z) = (xy)_(xz) ,(x_y)=x^y;
(c10)xyx^y ,xx= 0 ;
(c11)yx!y ,xx!y ;
(c12)x_x= 1)x^x= 0 ;
(c13)x_y((x!y)!y)^((y!x)!x) .
2

Prop osition 2.2. ([12],[20]) In e ach r e gular r esiduate d lattic e L , the fol lowing pr op erties hold for
al lx;y2L :
(i)x!y=y!x;
(ii)xy= (x!y).
Prop osition 2.3. ([20]) L etL b e a c ommutative r esiduate d lattic e, FL ,andx;y;z2L , then the
fol lowing assertions hold for al l x;y;z2L :
(c14)x;y2F)xy2F ;
(c15)x2F;xy)y2F ;
(c16) 12F ;
(c17)x;x!y2F)y2F ;
(c18)z;z!((x!y)!x)2F)x2F ;
(c19)x_x2F ;
(c20)x!(z!y);y!z2F)x!z2F ;
(c21)z!(x!y);z!x2F)z!y2F ;
(c22)x2!y2F)x!y2F ;
(c23)x!x22F ;
(c24)z;z!(y!x)2F)((x!y)!y)!x2F ;
(c25)y!x2F)((x!y)!y)!x2F ;
(c26) ((x!y)!y)!((y!x)!x)2F .
W e no w giv e the denition and some prop erties for the non-comm utativ e residuated lattice.
Denition 2.2. ([9],[10],[11],[13]) A non-c ommutative r esiduate d lattic e is a structur e L= (L;^;_;;!
; ;0;1) of typ e (2;2;2;2;2;0;0) satisfying the fol lowing axioms:
(NCRL 1) (L;_;^;0;1) is a b ounde d lattic e;
(NCRL 2) (L;;1) is monoid with a unit element 1;
(NCRL 3) F or al lx;y;z2L , we have
xyz,xy!z,yx z:
By anRlmonoid w e mean a residuated lattice whic h satises the divisibility condition
(div) : (x!y)x=x^y=x(x y):
Prop osition 2.4. ([9],[13]) L et L b e a non-c ommutative r esiduate d lattic e. F or al l x;y;z2L , we
have
(nc1)xy,x!y= 1,x y= 1 ;
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(nc2)x(x y)y ;
(nc3) (x!y)xy ;
(nc4)x!(y z) =y (x!z) ;
(nc5)xy)xzyz;zyzy ;
(nc6)xy)z!xz!y;y!zx!z ;
(nc7)xy)z xz y;y zx z ;
(nc8)x!y(y!z) (x!z) ;
(nc9)x (y z)!(x z ;
(nc10)x!y(z!x)!(z!y) ;
(nc11)x y(z x) (z y) .
3 Filters in residuated lattices
In this section w e mak e an analogy b et w een dieren t t yp es of lter in case of comm utativ e and
non-comm utativ e residuated lattice.
Let recall the denition of lter in a comm utativ e residuated lattice.
Denition 3.1. ([16]) A nonempty subset F ofL is c al le d a lter of L if, for al lx;y2L , it
satises:
(CF 1)x;y2F)xy2F ;
(CF 2)x2F;xy)y2F .
Prop osition 3.1. L etF b e a subset of L . ThenT2F(L) if and only if it satises the c onditions
(c16) and(c17) , for al lx;y2L .
Theorem 3.1. ([17],[20]) L et F b e a subset of L and12F . Then the fol lowing assertions ar e
e quivalent, for al l x;y;z2L :
(i)F is a lter of L ;
(ii)x!y;y!z2F)x!z2F ;
(iii)x!y;xz2F)yz2F ;
(iv)x;y2F;xy!z)z2F .
A subsetFL is a lter of L if and only if, for all x;y2L ,F is a deductiv e system of L , that
is
(CDS 1) 12F ;
(CDS 2)x2F;x!y2F)y2F .
W e no w see the denition of lter in a non-comm utativ e residuated lattice.
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Denition 3.2. ([9],[10],[11],[13]) A subset F is c al le d a lter of a r esiduate d lattic e L if, for al l
x;y2L , it satises :
(NCF 1) 12F ;
(NCF 2)x;y2F)xy2F ;
(NCF 3)x2F;x!y2F)y2F ;
Denition 3.3. ([9],[19]) A subset FL ,F is a lter if and only if, for al l x;y2L , it is a
de ductive system, that is
(NCDS 1) 12F ;
(NCDS 2)x2F;x!y2F)y2F ;
(NCDS 3)x2F;x y2F)y2F .
Remark 3.1. The set of al l lters in L is denote d by F(L) .
It is easy to see that the second implication, from the non-comm utativ e case, is necessary and
compulsory to dene lters.
F urthermore, w e dene the equiv alence relation on a comm utativ e residuated lattice L .
Denition 3.4. ([20]) L etF b e a lter of L . W e dene the r elation F onL as fol lows:
xFy,x!y;y!x2F;8x;y2L:
It is ob viously that F is a congruence relation. Let L=F denote the set of the congruence classes
ofF , that means L=F :=f[x]Fjx2Lg , where [x]F:=fy2LjxFyg . Dene
;*;u;t onL=F
as follo ws:
[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, whic h is called the quotient r esiduate d
lattic e with r esp e ct to F .
F orw ards, w e see what happ ens in the non-comm utativ e case.
Denition 3.5. ([20]) A lter F , it is c al le d normal when x!y2F if and only if x y2F for
al lx;y2L . F or any normal lter F ofL , a r elationF onL dene d by
xFy,x!y;y!x2F( orx y;y x2F)
is a c ongruenc e on L and a quotient structur e L=F =fx=Fjx2Lg byF is a r esiduate d lattic e
(sinc e the class of al l non-c ommutative r esiduate d lattic es is a variety).
F or an y congruence  onL , w e dene a subset FL b yF=fx2Lj(x;1)2g that is a
normal lter.
Denition 3.6. ([19],[20]) A subset F of a c ommutative r esiduate d lattic e L is c al le d an implic ative
lter ofL if, for al lx;y2L , it satises:
(CIF 1) 12F ;
(CIF 2)z;z!((x!y)!x)2F)x2F ;
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Denition 3.7. ([10],[20]) A subset F ofL is c al le d a p ositive implic ative lter of L if, for al l
x;y;z2L , it satises:
(CPIF 1) 12F ;
(CPIF 2)z!(x!y);z!x2F)z!y2F .
Denition 3.8. ([19],[20]) A subset F ofL is c al le d a Bo ole an lter of L if, for al lx2L , it
satises:
(CBF 1) 12F
(CBF 1)x_x2F .
Prop osition 3.2. ([20]) L etF b e a subset of L . ThenF is an implic ative lter of L if and only if
F is a Bo ole an lter of L .
Theorem 3.2. ([20]) L etF b e a lter of L . Then the fol lowing assertions ar e e quivalent, for al l
x;y2L :
(i)F is a Bo ole an lter of L ;
(ii) (x!y)!x2F)x2F ;
(iii) (x!x)!x2F ;
(iv) The quotient r esiduate d lattic e L=F is a Bo ole an algebr a.
Theorem 3.3. ([20]) In any r esiduate d lattic e L , the fol lowing c onditions ar e e quivalent:
(i)L is a Bo ole an algebr a;
(ii) A ny lter of L is a Bo ole an lter of L ;
(iii)f1g is a Bo ole an lter of L ;
(iv) (x!y)!x=x , for al lx;y2L .
Denition 3.9. ([20]) L etL b e a non-c ommutative r esiduate d lattic e. A subset F ofL is c al le d an
implic ative lter if it satises :
(NCIF 1) 12F ;
(NCIF 2)x!(y!z)2F andx y2F)x!z2F ;
(NCIF 3)x (y z)2F andx!y2F)x z2F .
Theorem 3.4. ([9]) L etL b e a non-c ommutative r esiduate ds lattic e and F a lter ofL . ThenF
is an implic ative lter of L if and only if, for al l x2L ,x!x22F andx x22F .
Pr o of. Supp oseF is an implicativ e lter of L . Then, since x!(x!x2) = 12F andx x=
12F , w e obtain x!x22F . Similarly , since x (x x2) = 12F andx!x= 12F , then
x x22F . So that, the assumption is correct.
Con v ersely , w e supp ose that, for all x2L ,F satisesx!x22F andx x22F . Supp ose that
x!(y!z)2F andx 2F . W e kno w that x!x22F so w e ha v e (x!(y!z))x(x
y)!(x!(y!z))x2(x y)2F . Also, (x!(y!z))x2(x y) =f(x!(y!
6

xz))xgfx(x y)g(y!z)yz . W e obtain (x!(y!z))x2(x y)!z= 12F
and(x!(y!z))!((x(x y))!z) = (x!(y!z))x(x y)!z2F . Since
x!(y!z)2F , w e ha v ex(x y)!z2F . Con v ersely , since (x y)x!x(x y)2F ,
w e also ha v e (x y)x!z2F , whic h means (x y)!(x!z)2F and sox!z2F b y the
assumption x y2F . Hence, the assumption is correct, F is an implicativ e lter.
Theorem 3.5. ([9]) Every implic ative lter is normal in any r esiduate d lattic e.
Pr o of. LetF b e an implicativ e lter and x!y2F . Sincex(x!y) (x!y)x2F and
(x!y)x y= 12F b y(x!y)xy , w e obtain x(x!y) y= (x!y) (x y)2F .
W e kno w that x!y2F and it follo ws immediately that x y2F .
Denition 3.10. ([9],[13]) L et L b e a r esiduate d lattic e. A subset F ofL is c al le d a p ositive
implic ative lter if it satises :
(NCPIF 1) 12F ;
(NCPIF 2)x!((y!z) y)2F andx2F)y2F ;
(NCPIF 3)x ((y z)!y)2F andx2F)y2F .
Theorem 3.6. ([9]) Every p ositive implic ative lter of L is an implic ative lter.
Pr o of. F rom the denition of p ositiv e implicativ e lter w e kno w that x!((y!z) y)2F and
x2F)y2F . No w, w e tak e x= 1 ,y=x!x2andz=x2and obtain
1!(((x!x2)!x2) (x!x2) = ((x!x2)!x2) (x!x2)x!(x!x2) = 12F;
hencex!x22F , so thatF is an implicativ e lter.
Theorem 3.7. ([9],[10]) L et F b e a lter of a r esiduate d lattic e L . Then the fol lowing assertions
ar e e quivalent, for al l x;y;2L :
(i) F is a p ositive implic ative lter;
(ii) if(x!y) x2F thenx2F and if (x y)!x2F thenx2F ;
(iii) ((x!y) x)!x2F and((x y)!x) x2F ;
(iv) (x x)!x2F and(x!x) x2F;
(v)F is a b o ole an lter, that is, x_x2F andx_x2F .
Denition 3.11. ([9]) L etL b e a non-c ommutative r esiduate d lattic e. A subset F ofL is c al le d a
b o ole an lter if it satises :
(NCBF 1) (x!y)!x2F)x2F , for al lx;y2L ;
(NCBF 2) (x y) x2F)x2F , for al lx;y2L .
Denition 3.12. ([14],[16]) A subset F of a c ommutative r esiduate d lattic e L is c al le d a fantastic
lter ofL if, for al lx;y;z2L , it satises :
(CFF 1) 12F ;
(CFF 2)z;z!(y!x)2F)((x!y)!y)!x2F ;
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A fuzzy subset v in a comm utativ e residuated lattice L is called a fantastic fuzzy lter ofL if for
an yx;y;z2F ,
(CFFF 1)v(1)v(x) ;
(CFFF 2)v(z!(y!x))^v(z)v(((x!y)!y)!x) .
Prop osition 3.3. ([16]) Every fantastic fuzzy lter of a r esiduate d lattic e L is a fuzzy lter of L .
Pr o of. Letv b e a fan tastic fuzzy lter of L . Let2[0;1] andv6=; . Then 12v . Let
x;x!y2v , whic h means, v(x);v(x!y) . Thenv(x!(1!y)) =v(x!y) ,
hencev(x!(1!y))^v(x) , th us, b yv(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,
v is a fuzzy lter of L .
Theorem 3.8. ([16]) A lter F of a r esiduate d lattic e L is fantastic if and only if vF is a fantastic
fuzzy lter of L .
Pr o of. LetF b e a lter of L . Let us supp ose that F is fan tastic. Let vF(z!(y!x))^vF(z) = .
ThenvF(z!(y!x)) ==vF(z) , th usz!(y!x)2F;z2F , and hence ((x!y)!y)!
x2F , that means v(((x!y)!y)!x) = . Therefore w e get that vF is a fan tastic fuzzy lter
ofL .
Con v ersely , let vF b e a fan tastic fuzzy lter of L . Letz!(y!x)2F andz2F . Then
vF(z!(y!x)) ==vF(z) hencevF(((x!y)!y)!x) = , and therefore ((x!y)!y)!
x2F . That means, F is a fan tastic lter of L .
Theorem 3.9. ([14](The or em 4.2, 4.4.),[16]) L et F b e a lter of a r esiduate d lattic e L . Then the
fol lowing c onditions ar e e quivalent.
(i)F is a fantastic lter of L ;
(ii)y!x2F implies ((x!y)!y)!x2F for everyx;y2L ;
(iii)x!x2F for everyx2L ;
(iv)x!z2F andy!z2F implies ((x!y)!y)!z2F for everyx;y;z2L .
Theorem 3.10. [16] L etF b e a lter of a r esiduate d lattic e L . Then the fol lowing c onditions ar e
e quivalent,
(i)vF is 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 .
Remark 3.2. The fantastic lter wil l b e c al le d a normal (MP) lter.
Denition 3.13. ([20]) A subset F ofL is c al le d an MV-lter of L if it is a lter of L that satises
the c ondition (c41) for al lx;y2L .
Prop osition 3.4. ([20]) L etF b e a subset of L . ThenF is a fantastic lter of L if and only if F
is an MV-lter of L .
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Theorem 3.11. ([20]) L etF b e a lter of L . ThenF is an MV-lter of L if and only if it satises
the c ondition (c42) for al lx;y2L .
Pr o of. Supp ose that F is an MV-lter of L , andx;y2L . By Prop osition 2.1 w e can easily obtain
thatx!(x_y) = 12F and(x_y)!y=x!y . Therefore it follo ws from (c41) that
(((x_y)!y)!y)!(x_y)2F , that is, ((x!y)!y)!(x_y)2F . On the other hand,
sincex_y(y!x)!x , w e ha v e that (x_y)!((y!x)!x) = 12F . Th us, b y Theorem 3.1
it follo ws that ((x!y)!y)!((y!x)!x)2F , so the condition (c42) is satised.
Con v ersely , supp ose that F satises the condition (c42) , and letx;y2L suc h thaty!x2F .
Since (y!x)!(((x!y)!y)!x) = ((x!y)!y)!((y!x)!x)2F , it follo ws from
Prop osition 3.1 that ((x!y)!y)!x2F . By Denition 3.17, this sho ws that F is an MV-lter
ofL .
Theorem 3.12. ([20]) L etF b e a lter of L . ThenF is an MV-lter of L if and only if the quotient
r esiduate d lattic e L=F is an MV-algebr a.
LetA b e a residuated lattice, v a fuzzy set on L andx;y;z2L . W e en umerate some assertions
that will b e further necessary:
(c27)v(xy)v(x)^v(y) ;
(c28)xy)v(x)v(y) ;
(c29)v(1)v(x) ;
(c30)v(y)v(x!y)^v(x) ;
(c31)v(x)(v(z!((x!y)!x)))^v(z) ;
(c32)v(x_x) =v(1) ;
(c33)v(x!z)(v(x!(z!y)))^v(y!z) ;
(c34)v(x!(x!y))v(x!y) ;
(c35)v(x!(x!y)) =v(x!y) ;
(c36)v(x!z)(v(x!(y!z)))^v(x!y) ;
(c37)v(x!x2) =v(1) ;
(c38)v(((x!y)!y)!x)v(y!x) ;
(c39)v(((x!y)!y)!x)(v(z!(y!x)))^v(z) ;
(c40)v(((x!y)!y)!((y!x)!x)) =v(1) ;
(c41)xy!z)v(z)v(x)^v(y) .
Denition 3.14. A fuzzy set v ofL is c al le d a fuzzy lter of L if it satises the c onditions (c27)
and(c28) for al lx;y2L .
The set of al l fuzzy lters of L is denote d by FF(L).
Theorem 3.13. ([20]) A fuzzy set v ofL is a fuzzy lter of L if and only if it satises the c onditions
(c29) and(c30) for al lx;y2L .
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Theorem 3.14. ([20]) A fuzzy set v ofL is a fuzzy lter of L if and only if it satises the c ondition
(c41) .
Prop osition 3.5. ([20]) A fuzzy set v ofL is a fuzzy lter of L if and only if for any 2[0;1] ,
the level set v is either empty or a lter of L .
Denition 3.15. A fuzzy lter v is c al le d a fuzzy Bo ole an lter of L (briey, FB-lter), if it satises
the c ondition (c32) .
A fuzzy set v ofL is c al le d a fuzzy implic ative lter of L if it is a fuzzy lter of L that satises
the c ondition (c33) .
W e will call the fuzzy implicativ e lter in Denition 3.15 a fuzzy p ositive implic ative lter .
Prop osition 3.6. ([20]) L etv b e a fuzzy set on L . Thenv is an FB-lter of L if and only if v is
a fuzzy implic ative lter of L .
Theorem 3.15. ([20]) L etv b e a fuzzy lter of L . Then the fol lowing assertions ar e e quivalent,
for al lx;y;z2L :
(i)v is 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 3.1. ([20]) A fuzzy set v ofL is an FB-lter of L if and only if it satises the c onditions
(c29) and(c31) .
Pr o of. Using Theorem 3.15, the equiv alence (i),(v) is ob vious. W e only need to pro v e that if
v satises the conditions (c29) and(c31) thenv is a fuzzy lter of L . Puttingx=y in(c31) , w e
ha v e thatv(x)(v(z!((x!x)!x)))^v(z) = (v(z!(1!x)))^v(z) =v(z!x)^v(z) .
Thereforev satises the condition (c46) . This and the condition (c29) imply that v is a fuzzy lter
ofL .
Theorem 3.16. ([20]) L etv b e a fuzzy of L . Then the fol lowing assertions ar e e quivalent:
(i)v is an FB-lter of L ;
(ii)L=vv(1) is a Bo ole an algebr a;
(iii)L=v is Bo ole an algebr a.
Prop osition 3.7. ([20]) L etF b e a subset of L . ThenF is an implic ative lter of L if and only if
F is a Bo ole an lter of L .
Remark 3.3. The p ositive implic ative lter in Denition 3.7 is c al le d an implic ative lter.
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Denition 3.16. A subsetF ofL is c al le d a G-lter of L if it is a lter of L that satises
z!(x!y);z!x2F)z!y2F , for al lx;y2L .
Prop osition 3.8. ([20]) L etF b e a subset of L . ThenF is a p ositive implic ative lter of L if and
only ifF is a G-lter of L .
Denition 3.17. A subsetF ofL is c al le d an MV-lter of L if it is a lter of L that satises
y!x2F)((x!y)!y)!x2F , for al lx;y2L .
Prop osition 3.9. ([20]) L etF b e a subset of L . ThenF is a fantastic lter of L if and only if F
is an MV-lter of L .
Theorem 3.17. ([20]) L etF b e a lter of L . ThenF is an MV-lter of L if and only if it satises
((x!y)!y)!((y!x)!x)2F , for al lx;y2L .
Pr o of. Supp ose that F is an MV-lter of L , andx;y2L . By Prop osition 2.1 w e can easily obtain
thatx!(x_y) = 12F and(x_y)!y=x!y . Therefore it follo ws from (c25) that
(((x_y)!y)!y)!(x_y)2F , that is, ((x!y)!y)!(x_y)2F . On the other hand,
sincex_y(y!x)!x , w e ha v e that (x_y)!((y!x)!x) = 12F . Th us, b y Theorem 3.1
it follo ws that ((x!y)!y)!((y!x)!x)2F , so the condition (c26) is satised.
Con v ersely , supp ose that F satises the condition (c26) , and letx;y2L suc h thaty!x2F .
Since (y!x)!(((x!y)!y)!x) = ((x!y)!y)!((y!x)!x)2F , it follo ws from
Prop osition 3.1 that ((x!y)!y)!x2F . By Denition 3.17, this sho ws that F is an MV-lter
ofL .
Denition 3.18. ([9]) L etL b e a non-c ommutative r esiduate d lattic e. A subset F ofL is c al le d
fantastic lter if it satises :
(NCFF 1)y!x2F)((x!y) y)!x2F ;
(NCFF 2)y x2F)((x y)!y) 2F .
Theorem 3.18. ([9]) A lter F of a non-c ommutative r esiduate d lattic e L is a fantastic lter if
and only if
((x!y) y)!(x_y)2F and((x y)!y) (x_y)2F:
Pr o of. Sincey!(x_y) = 12F , it follo ws that ((x_y!y) y)!x_y2F . It follo ws
fromx_y!y= (x!y)^(y!y) =x!y that ((x!y) y)!x_y2F . The pro of of
((x y)!y) x_y2F is similarly .
Con v ersely , w e can assume that y!x2F . Then, since x_y!x=y!x2F and
((x!y) y)!x_y2F , w e obtain ((x!y) y)!x2F .
Theorem 3.19. ([9]) A lter F of a non-c ommutative r esiduate d lattic e L is a fantastic lter if
and only if
((x!y) y)!((y!x) x)2F and((x y)!y) ((y x)!x)2F:
Pr o of. It follo ws from x_y(y!x) x thatx_y!((y!x) x) = 12F . So that
((x!y) y)!x_y2F and w e can conclude that ((x!y) y)!((y!x) x)2F .
Similarly , w e can pro v e ((x y)!y) ((y x)!x)2F .
Con v ersely , making x=x_y w e ha v e ((x_y)!y) y)!((y!x_y) x_y)2F . Since
x_y!y=x!y andy!x_y= 1 , w e can conclude that ((x!y) y)!x_y2F , that is,
F is a fan tastic lter.
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