# The Lattice of  Intuitionistic Fuzzy Filters in Residuated Lattices

Authors: Zhen Ming Ma
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The notion of tip-extended pair of intuitionistic fuzzy filters is introduced by which it is proved that the set of all intuitionistic fuzzy filters in a residuated lattice forms a bounded distributive lattice.


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## Body

## 1. Introduction

Nowadays, it is generally accepted that in fuzzy logic the algebraic structure should be a residuated lattice which was introduced by Ward and Dilworth [1]. Some other logical algebras such as MTL-algebras [2], BL-algebras [3], MV-algebras [4], G-algebras, Π-algebras, and NM-algebras [2], which are also called R0-algebras [5], are all able to be considered particular classes of residuated lattices. (For details, see, e.g., [6].)Filters are an important tool to study these logical algebras and the completeness of the corresponding nonclassical logics. On the one hand, filters are closely related to congruence relations with which one can associate quotient algebras [7]; on the other hand, various filters correspond to various sets of provable formula [3, 4]. A filter is also called a deductive system in BL-algebras [8]. It has been widely investigated in residuated lattices [7, 9–11] and particular residuated lattices [2, 3, 6, 8, 12–15].Since Rosenfeld [16] applied the notion of fuzzy sets [17] to abstract algebra and introduced the notion of fuzzy subgroups, the literature of various fuzzy algebraic concepts has been growing very rapidly [18]. In particular, in [19–21], the notion of tip-extended pair of fuzzy sets was introduced to investigate the lattices of all fuzzy normal subgroups and L-ideals.The notion of fuzzy filters was introduced, and some properties of them were obtained [22–24]. Moreover, based on the notion of intuitionistic fuzzy sets (IFS) proposed by Atanassov [25], the concept of the intuitionistic fuzzy filter in BL-algebras was introduced in [26]. However, the study of residuated lattices from the point of lattice theory is less frequent.In this paper, the intuitionistic fuzzy filter theory in residuated lattices is developed. This paper is organized as follows: in Section2, some basic concepts and properties of intuitionistic fuzzy sets and intuitionistic fuzzy filters in residuated lattices are recalled. In Section 3, by introducing the notion of tip-extended pair of intuitionistic fuzzy filters, it is proved that the set of all intuitionistic fuzzy filters forms a bounded distributive lattice. The last section concludes this paper.


## 2. Preliminaries

The concepts of residuated lattices and intuitionistic fuzzy filters will be extensively used in the sequel. Therefore, we recall their definitions and summarize their main properties.LetU≠∅. A mapping f:U→[0,1] is called a fuzzy set [17]. Let f and g be fuzzy sets on U. Then tip-extended pair of f and g [19, 20] can be defined by
(1)fg(x)={f(x),x≠1,f(1)∨g(1),x=1,gf(x)={g(x),x≠1,g(1)∨f(1),x=1.LetuA,vA:U→[0,1] be two fuzzy sets satisfying 0≤uA(x)+vA(x)≤1 for all x∈U. Then A=(uA,vA) is called an intuitionistic fuzzy set [25] (or equivalently denoted by A={〈x,uA(x),vA(x)〉∣x∈U}). The family of all intuitionistic fuzzy sets on U will be denoted by IFS(U).Basic operations on intuitionistic fuzzy sets are defined in the following way.LetA,B∈IFS(U). One has
(2)A∩B=(uA∧uB,vA∨vB),A∪B=(uA∨uB,vA∧vB),A⊆BiffuA≤uB,vA≥vB,A⊇BiffA⊆B,A=BiffA⊇B,A⊆B.Definition 1 (see [3]).
Aresiduated lattice is an algebra L=(L,∧,∨,⊗,→,0,1) such that (L,∧,∨,0,1) is a bounded lattice with the least element 0 and the greatest element 1, (L,⊗,1) is a commutative monoid, and (⊗,→) forms an adjoint pair; that is, x⊗y≤z if and only if x≤y→z for all x,y,z∈L.A nonempty subsetF of L is called afilter of L if (i) ∀x,y∈F,x⊗y∈F; (ii) ∀x∈F,y∈L,x≤y implies y∈F or, equivalently, (iii) 1∈F; and (iv) ∀x∈F,y∈L,x→y∈F implies y∈F.The following alternative definitions of intuitionistic fuzzy filters were proved in [26], but they can be similarly verified in residuated lattices.Definition 2.
LetA∈IFS(L). Then A is called an intuitionistic fuzzy filter if (1)
uA(x)≤uA(1), vA(x)≥vA(1) for all x∈L;(2)
uA(x)∧uA(x→y)≤uA(y) for all x,y∈L;(3)
vA(x)∨vA(x→y)≥vA(y) for all x,y∈L.The set of all intuitionistic fuzzy filters on a residuated latticeL will be denoted by IFF(L).Theorem 3.
LetA∈IFS(L). Then A is an intuitionistic fuzzy filter if and only if x⊗y≤z implies uA(x)∧uA(y)≤uA(z) and vA(x)∨vA(y)≥vA(z) for all x,y,z∈L.Theorem 4.
LetA∈IFS(L). Then A is an intuitionistic fuzzy filter if and only if the following assertions hold: (1)
x≤y implies uA(x)≤uA(y) and vA(x)≥vA(y) for all x,y∈L;(2)
uA(x)∧uA(y)≤uA(x⊗y) and vA(x)∨vA(y)≥vA(x⊗y) for all x,y∈L.


## 3. Lattice of Intuitionistic Fuzzy Filters

In this section, we mainly investigate the lattice of all intuitionistic fuzzy filters by introducing the notion of tip-extended pair of intuitionistic fuzzy sets.The following lemma is obvious but necessary.Lemma 5.
LetA,B be intuitionistic fuzzy filters of L. Then so is A∩B.ForA∈IFS(L), the intersection of all intuitionistic fuzzy filters containing A is called the generated intuitionistic fuzzy filter by A, denoted as 〈A〉.Theorem 6.
LetA∈IFS(L). Define a new intuitionistic fuzzy set B by B=(uB,vB) where
(3)uB(x)=⋁a1⊗⋯⊗an≤xai∈L,n∈N{uA(a1)∧⋯∧uA(an)},vB(x)=⋀a1⊗⋯⊗an≤xai∈L,n∈N{vA(a1)∨⋯∨uA(an)}
for all x∈L. Then B=〈A〉.Proof.
We complete the proof by two steps. Firstly, we verify thatB is an intuitionistic fuzzy filter. For all x,y∈L, such that x≤y, the definition of B yields that uB(x)≤uB(y) and vB(x)≥vB(y). For all x,y∈L, we have
(4)uB(x)∧uB(y)=⋁a1⊗⋯⊗an≤xai∈L,n∈N{uA(a1)∧⋯∧uA(an)}∧⋁b1⊗⋯⊗bm≤ybi∈L,m∈N{uA(b1)∧⋯∧uA(bm)}=⋁a1⊗⋯⊗an≤xai∈L,n∈N⋁b1⊗⋯⊗bm≤ybi∈L,m∈N{uA(a1)∧⋯∧uA(an)∧uA(b1)∧⋯∧uA(bm)}≤⋁c1⊗⋯⊗cn≤x⊗yci∈L,k∈N{uA(c1)∧⋯∧uA(ck)}=uB(x⊗y),vB(x)∨vB(y)=⋀a1⊗⋯⊗an≤xai∈L,n∈N{uA(a1)∨⋯∨uA(an)}∨⋀b1⊗⋯⊗bm≤ybi∈L,m∈N{uA(b1)∨⋯∨uA(bm)}=⋀a1⊗⋯⊗an≤xai∈L,n∈N⋀b1⊗⋯⊗bm≤ybi∈L,m∈N{uA(a1)∨⋯∨uA(an)∨uA(b1)∨⋯∨uA(bm)}≥⋀c1⊗⋯⊗cn≤x⊗yci∈L,k∈N{uA(c1)∨⋯∨uA(ck)}=uB(x⊗y).
Thus B is an intuitionistic fuzzy filter.
Secondly, letC be an intuitionistic fuzzy filter such that   C⊇A. By the definition of intuitionistic fuzzy filter, it holds that
(5)uB(x)=⋁a1⊗⋯⊗an≤xai∈L,n∈N{uA(a1)∧⋯∧uA(an)}≤⋁a1⊗⋯⊗an≤xai∈L,n∈N{uC(a1)∧⋯∧uC(an)}≤⋁a1⊗⋯⊗an≤xai∈L,n∈N{uC(a1⊗⋯⊗an)}≤uC(x),vB(x)=⋀a1⊗⋯⊗an≤xai∈L,n∈N{uA(a1)∨⋯∨uA(an)}≥⋀a1⊗⋯⊗an≤xai∈L,n∈N{uC(a1)∨⋯∨uC(an)}≥⋀a1⊗⋯⊗an≤xai∈L,n∈N{uC(a1⊗⋯⊗an)}≥uC(x),
and hence B⊆C. Thus B=〈A〉.Example 7.
LetL={0,a,b,1} with 0<a<b<1. The operations ⊗ and → are defined as
(6)⊗0ab100000a00aab0abb10ab1→0ab101111aa111b0a1110ab1
Define uA,vA:L→[0,1] as uA(0)=0.2,vA(0)=0.7,uA(a)=0.5,vA(a)=0.45,uA(b)=0.6,vA(b)=0.4,uA(1)=0.8, and vA(1)=0.15. Since 0.5=uA(a)∧uA(a)≮uA(a2)=0.2,f is not a fuzzy filter. It is routine to verify that 〈A〉 is an intuitionistic fuzzy filter, where u〈A〉(0)=u〈A〉(a)=0.5,v〈A〉(0)=v〈A〉(a)=0.4,u〈A〉(b)=0.6,v〈A〉(b)=0.3,u〈A〉(1)=0.8, and v〈A〉(1)=0.15 from the above theorem.Lemma 8.
Leta,b,s,t∈[0,1] such that 0≤a+b≤1 and 0≤s+t≤1. Then 0≤a∨s+b∧t≤1.Proof.
Not losing the generality, we assume thata≤s. Then a∨s+b∧t≤s+t≤1. It is obvious that 0≤a∨s+b∧t. Thus it holds that 0≤a∨s+b∧t≤1.Theorem 9.
LetA be an intuitionistic fuzzy filter of L and for all t,s∈[0,1] such that 0≤t+s≤1. Then At,s=(uAt,vAs) is an intuitionistic fuzzy filter, where
(7)uAt(x)={uA(x),x≠1,uA(1)∨t,x=1,vAs(x)={vA(x),x≠1,vA(1)∧s,x=1.Proof.
It follows from Lemma8 that At,s∈IFS(L). Now we prove that At,s is an intuitionistic fuzzy filter.
Ifx≤y, we consider the following two cases:
Case
1 (y=1). It is obvious that uAt(x)≤uAt(1)=uAt(y), vAt(x)≥vAt(1)=vAt(y).
Case
2 (y≠1). The definition of At,s leads that uAt(x)=uA(x)≤uA(y)=uAt(y), vAt(x)=vA(x)≥vA(y)=vAt(y).
ThusuAt(x)≤uAt(y), vAt(x)≥vAt(y).
Letx,y∈L. We consider the following two cases:
Case
1 (x⊗y=1). If x=y=1, it is obvious that uAt(x)∧uAt(y)≤uAt(x⊗y),vAt(x)∨yAt(x)≥vAt(x⊗y).
Ifx=1,y≠1 or x≠1,y=1, it is a contradiction.
Ifx≠1,y≠1, it holds that uAt(x)∧uAt(y)=uA(x)∧uA(y)≤uA(x⊗y)   =  uAt(x⊗y),vAt(x)∨yAt(x)   =  vA(x)∨yA(x)≥vA(x⊗y)=vAt(x⊗y).
Case
2 (x⊗y≠1). If x=y=1, it is a contradiction.
Ifx=1,y≠1 or x≠1,y=1, it is obvious that uAt(x)∧uAt(y)≤uAt(x⊗y),vAt(x)∨yAt(x)≥vAt(x⊗y).
Ifx≠1,y≠1,we have uAt(x)∧uAt(y)=uA(x)∧uA(y)≤uA(x⊗y)   =  uAt(x⊗y),vAt(x)∨yAt(x)   =     vA(x)∨yA(x)≥vA(x⊗y)   =    vAt(x⊗y).
All in all, it yields thatuAt(x)∧uAt(y)   =  uA(x)∧uA(y)≤uA(x⊗y)   =    uAt(x⊗y),vAt(x)∨yAt(x)   =     vA(x)∨yA(x)≥vA(x⊗y)   =   vAt(x⊗y).
ThusAt,s is an intuitionistic fuzzy filter.For givenA,B∈IFS(L), the operation ⋆~ is defined by
(8)A⋆~B=(uA⊗~uB,vA⊙~vB),
where
(9)uA⊗~uB(x)=⋁y⊗z≤x{uA(y)∧uB(z)},vA⊙~vB(x)=⋀y⊗z≤x{vA(y)∨vB(z)}.Furthermore, the tip-extended pair for intuitionistic fuzzy setsA and B are defined by
(10)AB=(uAuB,vAvB),
where
(11)uAuB(x)={uA(x),x≠1,uA(1)∨uB(1),x=1,vAvB(x)={vA(x),x≠1,vA(1)∧vB(1),x=1,BA=(uBuA,vBvA),
where
(12)uBuA(x)={uB(x),x≠1,uB(1)∨uA(1),x=1,vBvA(x)={vB(x),x≠1,vB(1)∧vA(1),x=1.Theorem 10.
LetA,B∈IFF(L). Then AB⋆~BA∈IFF(L).Proof.
It is obvious thatuAuB⊗~uBuA is order-preserving, and vAvB⊙~vBvA is order-reserving. For all x,y∈L, we have
(13)uAuB⊗~uBuA(x⊗y)=⋁u⊗v≤x⊗y{uAuB(u)∧uBuA(v)}≥⋁p⊗q≤xr⊗s≤y{uAuB(p⊗r)∧uBuA(q⊗s)}≥⋁p⊗q≤xr⊗s≤y{uAuB(p)∧uAuB(r)∧uBuA(q)∧uBuA(s)}=⋁p⊗q≤x{uAuB(p)∧uBuA(q)}∧⋁r⊗s≤y{uAuB(r)∧uBuA(s)}=uAuB⊗~uBuA(x)∧uAuB⊗~uBuA(y),vAvB⊙~vBvA(x⊗y)=⋀u⊗v≤x⊗y{vAvB(u)∨vBvA(v)}≤⋀p⊗q≤xr⊗s≤y{vAvB(p⊗r)∨vBvA(q⊗s)}≤⋀p⊗q≤xr⊗s≤y{vAvB(p)∨vAvB(r)∨vBvA(q)∨vBvA(s)}=⋀p⊗q≤x{vAvB(p)∨vBvA(q)}∨⋀r⊗s≤y{vAvB(r)∨vBvA(s)}=vAvB⊙~vBvA(x)∨vAvB⊙~vBvA(y),
and hence uAuB⊗~uBuA(x⊗y)≥uAuB⊗~uBuA(x)∧uAuB⊗~uBuA(y),vAvB⊙~vBvA(x⊗y)≤vAvB⊙~vBvA(x)∨vAvB⊙~vBvA(y).
ThusAB⋆~BA∈IFF(L).Theorem 11.
LetA,B∈IFF(L). Then AB⋆~BA=〈A∪B〉.Proof.
It is easy to prove thatA,B⊆AB⋆~BA, and hence A∪B⊆AB⋆~BA. Thus 〈A∪B〉⊆AB⋆~BA.
Assume thatC∈IFF(L) such that A∪B⊆C. If x=1, we have uAuB⊗~uBuA(1)=uA(1)∨uB(1)≤uC(1),vAvB⊙~vBvA(1)=vA(1)∧vB(1)≥vC(1). If x<1, it holds that
(14)uAuB⊗~uBuA(x)=⋁y⊗z≤x{uAuB(y)∧uBuA(z)}=⋁y⊗z≤xy≠1,z≠1{uAuB(y)∧uBuA(z)}∨⋁y≤x{uA(y)}∨⋁z≤x{uB(z)}=⋁y⊗z≤xy≠1,z≠1{uA(y)∧uB(z)}∨⋁y≤x{uA(y)}∨⋁z≤x{uB(z)}≤⋁y⊗z≤xy≠1,z≠1{uC(y)∧uC(z)}∨⋁y≤x{uC(y)}∨⋁z≤x{uC(z)}=⋁y⊗z≤x{uA(y)∧uB(z)}≤uC(x),vAvB⊙~vBvA(x)=⋀y⊗z≤x{vAvB(y)∨vBvA(z)}=⋀y⊗z≤xy≠1,z≠1{vAvB(y)∨vBvA(z)}∧⋀y≤x{vA(y)}∧⋀z≤x{vB(z)}=⋀y⊗z≤xy≠1,z≠1{vA(y)∨vB(z)}∧⋀y≤x{vA(y)}∧⋀z≤x{vB(z)}≥⋀y⊗z≤xy≠1,z≠1{vC(y)∨vC(z)}∧⋀y≤x{vC(y)}∧⋀z≤x{vC(z)}=⋀y⊗z≤x{vA(y)∨vB(z)}≥vC(x).
It follows from Theorem 10 that AB⋆~BA=〈A∪B〉.Associating with the above results, we prove the main theorem here. ForA,B∈IFF(L), the operations ⊓ and ⊔ on IFF(L) are defined by
(15)A⊓B=A∩B,A⊔B=AB⋆~BA.Theorem 12.
(
IFF
(
L
)
,
⊓
,
⊔
,
∅
,
L
) is a bounded distributive lattice.Proof.
We only verify the distributivity. Obviously, it holds thatC⊓(A⊔B)⊇(C⊓A)⊔(C⊓B), so we only prove that C⊓(A⊔B)⊆(C⊓A)⊔(C⊓B). Assume that x∈L for uC∧uAuB⊗~uBuA(x)≤(uC∧uA)uC∧uB⊗~(uC∧uB)uC∧uA(x), we consider the following two cases.
Case
1 (x=1). We have
(16)uC∧uA⊔B(1)=uC(1)∧uAuB⊗~uBuA(1)=uC(1)∧(uA(1)∨uB(1))=(uC(1)∧uA(1))∨(uC(1)∧uB(1))=(uC∧uA)uC∧uB⊗~(uC∧uB)uC∧uA(1),vC∨vA⊔B(1)=vC(1)∨vAvB⊙~vBvA(1)=vC(1)∨(vA(1)∧vB(1))=(vC(1)∨vA(1))∧(vC(1)∨vB(1))=(vC∨vA)vC∨vB⊙~(vC∨vB)vC∨vA(1);
Case
2 (x≠1). It holds that
(17)uC(x)∧uAuB⊗~uBuA(x)=uC(x)∧⋁y⊗z≤x{uAuB(y)∧uBuA(z)}=⋁y⊗z≤x{uC(x)∧uAuB(y)∧uBuA(z)}=⋁y⊗z≤xy≠1,z≠1{uC(x)∧uA(y)∧uC(x)∧uB(z)}∨{uC(x)∧uAuB(1)∧uB(x)}∨{uC(x)∧uA(x)∧uBuA(1)}=⋁y⊗z≤xy≠1,z≠1{uC(x)∧uA(y)∧uC(x)∧uB(z)}∨{uC(x)∧uC(1)∧uAuB(1)∧uB(x)}∨{uC(x)∧uC(1)∧uA(x)∧uBuA(1)}=⋁y⊗z≤xy≠1,z≠1{uC(x)∧uA(y)∧uC(x)∧uB(z)}∨{uC(1)∧(uA(1)∨uB(1))n∧[(uC(x)∧uB(x))∨(uC(x)∧uA(x))]}=⋁y⊗z≤xy≠1,z≠1{uC(x)∧uA(y)∧uC(x)∧uB(z)}∨{[(uC(1)∧uA(1))∨(uC(1)∧uB(1))]n∧[(uC(x)∧uB(x))∨(uC(x)∧uA(x))]}≤⋁y⊗z≤xy≠1,z≠1{(uC∧uA)uC∧uB(y∨x)nl∧(uC∧uB)uC∧uA(z∨x)}∨[(uC∧uA)uC∧uB(1)∧(uC∧uA)(y∨x)]∨[(uC∧uB)uC∧uA(1)∧(uC∧uB)(z∨x)]=⋁y⊗z≤x{(uC∧uA)uC∧uB(y∨x)nn∧(uC∧uB)uC∧uA(z∨x)}.
Let y∨x=y′ and z∨x=z′; it is easy to verify that y′⊗z′≤x, and then the above can be written as
(18)=⋁y′⊗z′≤x{(uC∧uA)uC∧uB(y′)∧(uC∧uB)uC∧uA(z′)}=(uC∧uA)uC∧uB⊗~(uC∧uB)uC∧uA(x),vC(x)∨vAvB⊙~vBvA(x)=vC(x)∨⋀y⊗z≤x{vAvB(y)∨vBvA(z)}=⋀y⊗z≤x{vC(x)∨vAvB(y)∨vBvA(z)}=⋀y⊗z≤xy≠1,z≠1{vC(x)∨vA(y)∨vC(x)∨vB(z)}∧{vC(x)∨vAvB(1)∨vB(x)}∧{vC(x)∧vA(x)∧vBvA(1)}=⋀y⊗z≤xy≠1,z≠1{vC(x)∨vA(y)∨vC(x)∨vB(z)}∧{vC(x)∨vC(1)∨vAvB(1)∨vB(x)}∧{vC(x)∨vC(1)∨vA(x)∨vBvA(1)}=⋀y⊗z≤xy≠1,z≠1{vC(x)∨vA(y)∨vC(x)∨vB(z)}∧{vC(1)∨(vA(1)∧vB(1))m∨[(vC(x)∨vB(x))∧(vC(x)∨vA(x))]}=⋀y⊗z≤xy≠1,z≠1{vC(x)∨vA(y)∨vC(x)∨vB(z)}∧{[(vC(1)∨vA(1))∧(vC(1)∨vB(1))]n∨[(vC(x)∨vB(x))∧(vC(x)∨vA(x))]}≥⋀y⊗z≤xy≠1,z≠1{(vC∨vA)vC∨vB(y∨x)n∨(vC∨vB)vC∨vA(z∨x)}∧[(vC∨vA)vC∨vB(1)∨(vC∨vA)(y∨x)]∧[(vC∨vB)vC∨vA(1)∧(vC∨vB)(z∨x)]=⋀y⊗z≤x{(vC∨vA)vC∨vB(y∨x)nnnnnnnnnnn∨(vC∨vB)vC∨vA(z∨x)}.
Let y∨x=y′ and z∨x=z′; it is easy to verify that y′⊗z′≤x, and then the above can be written as
(19)=⋀y′⊗z′≤x{(vC∨vA)vC∨vB(y′)∨(vC∨vB)vC∨vA(z′)}=(vC∨vA)vC∨vB⊙~(vC∨vB)vC∨vA(x).
ThusuC∧uAuB⊗~uBuA≤(uC∧uA)uC∧uB⊗~(uC∧uB)uC∧uA and vC(x)∨vAvB⊙~vBvA(x)≥(vC∨vA)vC∨vB⊙~(vC∨vB)vC∨vA; that is, the distributivity holds.


## 4. Conclusions

In this paper, by the notion of tip-extended pair of intuitionistic fuzzy sets, it is verified that the set of all intuitionistic fuzzy filters forms a bounded distributive lattice.Future research will focus on the lattice of fuzzy notions in other algebras by tip-extended pair.



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