# ℒ-Fuzzy Cosets of ℒ-Fuzzy Filters of Residuated Multilattices

Authors: Pierre Carole Kengne; Blaise Bleriot Koguep Njionou; Daquin Cèdric Awouafack; Luc Eméry Diékouam Fotso
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

This paper mainly focuses on building theℒ-fuzzy filter theory of residuated multilattices. Firstly, we introduce the concepts of ℒ-fuzzy filter and ℒ-fuzzy deductive system of residuated multilattices. Then, we highlight their properties and show how they are linked. Secondly, we introduce the concept of prime ℒ-fuzzy filter and propose some illustrative examples. Then, we bring out their properties and show how they are related to the concept of ℒ-fuzzy prime filter. Thirdly, we characterize ℒ-fuzzy maximal filter and maximal ℒ-fuzzy filter by atoms and coatoms. In the case where ℒ is a distributive lattice, we prove that maximal ℒ-fuzzy filters are prime. Finally, we are interested in ℒ-fuzzy cosets of an ℒ-fuzzy filter, and we prove that the set of all ℒ-fuzzy cosets of any ℒ-fuzzy filter of a residuated multilattice is a residuated multilattice.


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## Body

## 1. Introduction

The concepts of ordered and algebraic multilattice were introduced by Benado [1]. A multilattice is an algebraic structure in which the restrictions imposed on a complete lattice, namely, the “existence of least upper bounds and greatest lower bounds,” are relaxed to the “existence of minimal upper bounds and maximal lower bounds.” Cabrera et al. [2] introduced the notion of residuated multilattice as a partially ordered commutative residuated integral monoid (briefly pocrim) whose poset is a multilattice [3]. Therefore, residuated multilattices generalize both residuated lattices and multilattices.Also, in 1965, Zadeh [4] introduced the notion of fuzzy subset of a nonempty set X as a function from X into the unit interval 0,1 to describe, study, and formulate mathematically those objects which are not well defined. Many authors have investigated fuzzy algebraic notions taking the linearly ordered set 0,1 to be the set of degrees of membership. However, as Goguen [5] pointed out, in some situations, the structure of a complete bounded lattice ℒ can be a suitable set of truth values.In the same way, Tonga [6] studied the notion of maximal ℒ-fuzzy filters and ℒ-fuzzy maximal filters of bounded lattices. He showed that any maximal ℒ-fuzzy is also an ℒ-fuzzy maximal and gave characterizations of these. Following him, Kadji et al. [7] introduced the notions of ℒ-fuzzy filter and ℒ-fuzzy prime (and maximal) filter of residuated lattices and gave some characterizations of these notions. However, these different notions have not yet been explored in the residuated multilattice. In this paper, we introduce the notion of ℒ-fuzzy filter and ℒ-fuzzy coset of an ℒ-fuzzy filter in residuated multilattices.The paper is organized as follows: In Section1, we recall some basic notions on residuated multilattice and fix the different notations. In Section 2, we study the notion of ℒ-fuzzy filters with some examples and their properties. Section 3 is devoted to the notions of ℒ-fuzzy prime filter and prime ℒ-fuzzy filter. We give a complete characterization and several properties of ℒ-fuzzy prime filter and prime ℒ-fuzzy filter. In Section 4, we study ℒ-fuzzy maximal filters and maximal ℒ-fuzzy filters, and the relationships existing between these different classes of ℒ-fuzzy filters will also be given. In the last section, we will study the notion of cosets of a ℒ-fuzzy filter; we have shown that the set of cosets of a ℒ-fuzzy filter is also a residuated multilattice.


## 2. Preliminaries and Notations

In this preliminary section, we recall some essential facts about filters, congruences, and homomorphism of residuated multilattices; the reader is expected to refer to [2, 8–10] or [11] for details.GivenM,⊑ a partially ordered set and A⊆M, a multisupremum of A is a minimal element of the set of upper bounds of A, and multisupA denotes the set of multisuprema of A. Dually, we define the multi-infima, which will be denoted as multinfA.Definition 1 (see [12]).
A posetM,⊑ is an ordered multilattice if it satisfies that, for all a,b,x∈M with a⊑x and b⊑x, there exist z∈multisupa,b such that z⊑x and its dual version for multinfa,b. A multilattice is said to be full if multisupa,b≠∅ and multinfa,b≠∅, for all a,b∈M.
We will writea⊔b to denote multisupa,b and a⊓b to denote the set multinfa,b.
In the following, we recall the comparability properties, which have an important role in multilattice theory.Proposition 1 (see [12]).
In any full multilatticeM;⊑,⊓,⊔, the following properties (called comparability laws) are satisfied:(C1)
c∈a⊔b implies that a⊑c and b⊑c.(C2)
c∈a⊓b implies that a⊒c and b⊒c.(C3)
(c,d∈a∗b and c⊑d) implies that c=d, where ∗∈⊔,⊓.Proposition 2.
In a full multilatticeM;⊑,⊓,⊔, we have that, for all a,b,x,y∈M,(i)
ifx,y∈a⊔b and x≠y, then there exists a′,b′∈x⊓y such that x,y∈a′⊔b′;(ii)
ifx,y∈a⊓b and x≠y, then there exists a′,b′∈x⊔y such that x,y∈a′⊓b′.Proof.
(i)
Letx,y∈a⊔b such that x≠y. We have a,b⊑x,y, and then there exists a′,b′∈x⊓y such that a⊑a′ and b⊑b′. Let z,z′∈M such that a′,b′⊑z,z′, z⊑x, and z′⊑y. We have a⊑a′⊑z,z′ and b⊑b′⊑z,z′; then, a,b⊑z,z′. Therefore, x=z and y=z′ because x,y∈a⊔b. Thus, x,y∈a′⊔b′.(ii)
The proof is similar toi.Lemma 1.
LetM;⊑,⊓,⊔ be a full multilattice and a,b,c, three elements in M. For all x∈a⊓c and y∈b⊓c, if x⊑y, then x∈a⊓y.Proof.
Leta,b,c∈M such that x∈a⊓c and y∈b⊓c. Suppose that x⊑y. We have x⊑a,y. Let z∈M such that z⊑a,y and x⊑z. We have z⊑a,y and y⊑c. Then, z⊑a,c. Since x⊑z and x∈a⊓c, we obtain x=z. Thus, x∈a⊓y.Definition 2 (see [2]).
A tupleA=A;⊑,⊙,⟶,⊤ is said to be a partially ordered commutative residuated integral monoid, briefly a pocrim, if the following properties hold:(1)
A,⊙,⊤ is a commutative monoid with neutral element ⊤.(2)
A,⊑ is a poset with maximum ⊤.(3)
The operations⊙ and ⟶ satisfy the adjointness condition; that is, a⊙c⊑b if and only if c⊑a⟶b, for every a,b,c∈A.
We now recall the following useful conditions that hold in a pocrimA, for all x,y,z∈A, which will be used:(P1)
x⊙y⊑x,y (i.e., x⊑y⟶x and y⊑x⟶y).(P2)
x⊙x⟶y⊑x⊑y⟶x⊙y and x⊙x⟶y⊑y⊑x⟶x⊙y.(P3)
ifx⊑y, then x⊙z⊑y⊙z, z⟶x⊑z⟶y, and y⟶z⊑x⟶z.(P4)
x⟶y⟶z=y⟶x⟶z.(P5)
x⟶y⊙y⟶z⊑x⟶z.(P6)
x⟶y⊑x⊙z⟶y⊙z.(P7)
x⟶y⊑z⟶x⟶z⟶y and x⟶y⊑y⟶z⟶x⟶z.Definition 3 (see [2]).
A residuated multilattice is a pocrim whose underlying poset is a multilattice. If, in addition, there exists a bottom element, the residuated multilattice is said to be bounded.Example 1.
LetM,⊑ be the multilattice described in Figure 1.
Let the subsetsA=ai,0≤i≤4 and B=bi,0≤i≤2. We define ⊙ and ⟶ as follows:(1)x⊙y=⊥,ifx=⊥ory=⊥orx=a0andy∈Aorx∈Aandy=a0,a1,ifx,y∈A\a0,x,ify=⊤orx∈Aandy∈B,y,ifx=⊤orx∈Bandy∈A,b1,ifx=b1andy=b1,b2,ifx=b2andy=b2,b0,otherwise,
and(2)x⟶y=⊤,ifx≤y,b2,ifx=b1andy∈b0,b2,b1,ifx=b2andy∈b0,b1,y,ifx=⊤orx∈Bandy∈A∪⊥,a0,ifx∈A\a0andy∈⊥,a0,a4,otherwise.
It is a routine calculation thatℳ=M;⊑,⊙,⟶,⊤,⊥ is a bounded residuated multilattice.Figure 1
Hasse diagram ofM.Lemma 2 (see [2]).
Every residuated multilattice is full.
Given residuated multilatticeℳ=M;⊑,⊙,⟶,⊤,⊥, ⊙,⟶,⊓, and ⊔ can be extended to P∗M as follows: for all A,B∈P∗M, A⊙B=a⊙b|a∈Aandb∈B, A⟶B=a⟶b|a∈Aandb∈B, A⊓B=∪a∈A,b∈Ba⊓b, and A⊔B=∪a∈A,b∈Ba⊔b. Particularly, for all z∈M, A⊙z,A⟶z,z⟶A,A⊓z, and A⊔z will stand for A⊙z,A⟶z,z⟶A,A⊓z, and A⊔z, respectively.Proposition 3 (see [2]).
In a residuated multilatticeℳ=M;⊑,⊙,⟶,⊤,⊥, the following inclusions hold for all a,b,c∈M:(i)
a⟶c⊓b⟶c⊆a⊔b⟶c.(ii)
c⟶a⊓c⟶b⊆c⟶a⊓b.
Particularly,a⟶b∈a⊔b⟶b and a⟶b∈a⟶a⊓b.
LetM,⊑ be a poset; the set of upper bounds of x∈M will be denoted by x↑=a∈M,a⊒x, and dually x↓=a∈M,a⊑x denotes the set of lower bounds. Similarly, for X⊆M, upper closure isupper closure isX↑=∪x∈Xx↑, and lower closure is X↓=∪x∈Xx↓.Proposition 4 (see [2]).
[2] In a residuated multilattice ℳ=M;⊑,⊙,⟶,⊤,⊥, the following conditions hold for all x,y,z∈M:(1)
x⊙y,x⊙x⟶y∈x⊓y↓.(2)
x⊙y⊓z⊆x⊙y⊓x⊙z↓.(3)
x⊙y⊔z⊆x⊙y⊔x⊙z↑.(4)
x⊙y⊔x⊙z⊆x⊙y⊔z.Definition 4.
A complete latticeL;∧,∨ is said to be a completely meet-distributive lattice if, for all a∈L and any family bj,j∈I of L, we have a∧∨j∈Ibj=∨j∈Ia∧bj.Definition 5.
LetL,≤ be a lattice and α∈L.(i)
α is said to be ∧-prime if α≠1, and for any β,γ∈L, β∧γ≤α implies either β≤α or γ≤α.(ii)
α is said to be an atom of L if there is no β∈L such that 0<β<α.(iii)
α is said to be a coatom of L if there is no β∈L such that α<β<1.
Due to the combination of two structures, multilattice and pocrim, for any residuated multilattice, we have the notion of filter of pocrim (p-filter), the notion of filter of multilattice (m-filter), and the notion of filter of residuated multilattice (filter).Definition 6 (see [2]).
Given a pocrim, a non empty subsetF⊆A is a p-filter of A if the following conditions hold:(i)
Ifa,b∈F, then a⊙b∈F.(ii)
Ifa⊑b and a∈F, then b∈F.
F is a deductive system if it satisfies the following:(1)
⊤∈F.(2)
a⟶b∈F and a∈F imply b∈F, for all a,b∈A.Proposition 5.
In the pocrim, the notions of p-filters and deductive systems are equivalent.Proof.
LetA=A;⊑,⊙,⟶,⊥,⊤ be a pocrim and ∅≠F⊆A.⇒) Suppose that F is a p-filter. We have F≠∅; then, there exists x∈A such that x∈F. Moreover, x⊑⊤; therefore, ⊤∈F.Leta,b∈A such that a⟶b∈F and a∈F. Then, we havea⟶b⊑a⟶b. Therefore, b∈F.Thus,F is a deductive system.⇐) Suppose that F is a deductive system. Let a,b∈M such that a⊑b and a∈F. We have a⟶b=⊤∈F and a∈F,then b∈F by hypothesis.
Leta,b∈F. Since a∈F, therefore, a⊙b∈F.
ThusF is a p-filter.Definition 7 (see [8]).
LetM;⊔,⊓ be a multilattice. A nonempty set F⊆M is said to be an m-filter if the following conditions hold: for all a,b∈M,(M1)
a,b∈F implies ∅≠a⊓b⊆F(M2)
a∈F implies a⊔x⊆F, for all x∈M(M3)
Ifa⊔b∩F≠∅, then a⊔b⊆F.Definition 8 (see [2]).
Letℳ=M;⊑,⊙,⟶,⊤,⊥ be a residuated multilattice. A nonempty set F⊆M is said to be filter, if it is a deductive system and the following condition holds: a⟶b∈F implies a⊔b⟶b⊆F and a⟶a⊓b⊆F.
A filterF of a residuated multilattice M;⊔,⊓,⊙,⟶,⊤,⊥ is said to be proper if F≠M; it is non trivial if⊤⊈F⊈M.
The existing relation between filter and m-filter is given by the following theorem.Theorem 1 (see [2]).
Letℳ=M;⊑,⊙,⟶,⊤,⊥ be a residuated multilattice and F be a deductive system of ℳ; F is a filter if and only if(1)
F is an m-filter for all a,b∈M(2)
for allx,y∈a⊔b, if x⟶y∈F, then y⟶x∈F(3)
for allx,y∈a⊓b, if x⟶y∈F, then y⟶x∈FDefinition 9 (see [2]).
LetM;⊔,⊓,⊙,⟶,⊤,⊥ be a residuated multilattice. A deductive system F is said to be consistent if, for all a,b,c∈M, the following conditions hold:(i)
Ifa⟶c,b⟶c∈F, then a⊔b⟶c⊆F.(ii)
Ifc⟶a,c⟶b∈F, then c⟶a⊓b⊆F.Definition 10.
LetF be a proper filter of residuated multilattice ℳ=M;⊑,⊙,⟶,⊤,⊥.
F is a prime filter of ℳ if, for all x,y∈M, x⊔y⊂F implies that x∈F or y∈F.
Let∼ be a binary relation on M and X,Y⊆M; then,X∼^Y denotes that, for all x∈X, there exists y∈Y such that x∼y, and for all z∈Y, there exists t∈X such that t∼z.Definition 11 (see [8]).
LetM;⊓,⊔ be a multilattice; a congruence on M is any equivalence relation ∼ such that if a∼b, then a⊓c∼^b⊓c and a⊔c∼^b⊔c for all a,b,c∈M.Theorem 2 (see [8]).
LetM;⊓,⊔ be a multilattice and ∼ be a binary relation on M. Then, ∼ is a congruence relation if and only if the following conditions hold:(i)
∼ is reflexive.(ii)
x∼y if and only if there exist z∈x⊓y and ω∈x⊔y with z∼ω.(iii)
Ifx⊑y⊑z with x∼y and y∼z, then x∼z.(iv)
Ifx⊑y with x∼y, then x⊓t∼^y⊓t and x⊔t∼^y⊔t.Definition 12 (see [12]).
LetM;⊑,⊙,⟶ be a residuated multilattice; a congruence on M is any equivalence relation ∼ such that if a∼b, then a⊙c∼b⊙c; a⟶c∼b⟶c; c⟶a∼c⟶b; a⊓c∼^b⊓c; and a⊔c∼^b⊔c for all a,b,c∈M.
The concept of homomorphism between hyperstructures [13–15] has been considered from different points of view in the literature. The most used definition is that introduced originally by Benado [1] in the framework of multilattice.Definition 13 (see [8]).
A maph:M⟶M′ between multilattices is said to be a homomorphism if ha⊓b⊆ha⊓hb and ha⊔b⊆ha⊔hb for all a,b∈M.Definition 14 (see [2]).
Leth:M⟶M′ be a map between residuated multilattices; h is said to be a homomorphism if h is a multilattice homomorphism and also ha⊙b=ha⊙hb and ha⟶b=ha⟶hb for all a,b∈M.


## 3.ℒ-Fuzzy Filters of Residuated Multilattices

Letℳ=M;⊓,⊔,⊙,⟶,⊥,⊤ be a bounded residuated multilattice and ℒ=L;∧,∨,0,1 be a complete lattice. We will denote by ⊑ and ≤ the induced order relation defined on ℳ and ℒ, respectively.Definition 15 (see [6, 7]).
LetM be a nonempty set.(i)
A mapμ:M⟶L is called an ℒ-fuzzy subset of M.(ii)
Anℒ-fuzzy subset μ of M is called proper if it is not a constant map.(iii)
Ifμ:M⟶L is an ℒ-fuzzy subset of M and α∈L, then μα=a∈M;μa≥α is called the α-cut set of μ.
The notion ofℒ-fuzzy filter of a pocrim denoted by ℒ-fuzzy p-filter is defined as follows.Definition 16.
GivenA=A;⊑,⊙,⟶,⊥,⊤ a pocrim. Anℒ-fuzzy subset μ of A is an ℒ-fuzzy p-filter of A if the following conditions hold for all a,b∈A :(i)
Ifa⊑b, then μa≤μb.(ii)
μa⊙b=μa∧μb.Example 2.
LetE=α,β.PE;∩,∪,∅,E is a complete lattice.
LetA=0,a,b,c,d,1 with 0<a<d<1 and 0<b<c<d<1, but a is incomparable with b and c. The Hasse diagram of A is given in Figure 2.
Then,A=A;≤,⊙,⟶ is a pocrim with respect to the following operations (Tables 1 and 2).
The mapμ:A⟶PE defined by(3)μx=∅,ifx=0,α,ifx∈b,c,β,ifx=a,E,ifx∈d,1,is an PE-fuzzy p-filter of A. Since α and β are incomparable, then Imμ is not a chain.
The following result characterizes theℒ-fuzzy p-filter of pocrim.Figure 2
Hasse diagram ofA.Table 1
Cayley table of multiplication.
⊙0abcd10000000a0a00aab00bbbbc00bbbcd0abbdd10abcd1Table 2
Cayley table of implication.
⟶0abcd10111111ac1cc11baa1111caad111d0acc1110abcd1Proposition 6 (see [7]).
Letμ be an ℒ-fuzzy subset of a pocrim A=A;⊑,⊙,⟶,⊥,⊤. μ is an ℒ-fuzzy p-filter of A if and only if, for all α∈L, μα is either empty or p-filter of A.Definition 17.
GivenA=A;⊑,⊙,⟶,⊥,⊤ a pocrim, an ℒ-fuzzy subset μ of A is an ℒ-fuzzy deductive system of A if it satisfies the following:(1)
μ⊤≥μa, for all a∈A.(2)
μb≥μa∧μa⟶b, for all a,b∈A.
The following proposition shows that, in a pocrim, the notions ofℒ-fuzzy p-filter and ℒ-fuzzy deductive system are equivalent.Proposition 7 (see [7]).
LetA=A;⊑,⊙,⟶,⊥,⊤ be a bounded pocrim. An ℒ-fuzzy subset μ of A is an ℒ-fuzzy p-filter of A if and only if it is an ℒ-fuzzy deductive system.Definition 18.
LetM;⊓,⊔ be a multilattice. An ℒ-fuzzy subset μ of M is an ℒ-fuzzy m-filter if it satisfies the following conditions: for all a,b∈M,(FM1)
ifa⊑b, then μa≤μb;(FM2)
∀x∈a⊓b, μx=μa∧μb;(FM3)
∀x,y∈a⊔b, μx=μy.Example 3.
LetL=0,α,β,γ,δ,1 with 0<α<β<γ,δ<1, but γ and δ are incomparable. The Hasse diagram of L is given in Figure 3.
Then,ℒ=L,≤ is a complete lattice. Let us consider the multilattice M whose Hasse diagram is given in Figure 1. The map ν:M⟶L defined by(4)νx=0,ifx∈A\a4∪⊥,α,ifx=a4,β,ifx=b0,γ,ifx=b1,δ,ifx=b2,1,ifx=⊤,is an ℒ-fuzzy m-filter of M. We observe that Imν is not a chain because νb1=γ and νb2=δ are incomparable.Figure 3
Hasse diagram ofL.Remark 1.
Ifμ is an ℒ-fuzzy m-filter of a multilatticeM, then, for all x∈a⊔b, μx≥μa∨μb, for all a,b∈M. Indeed, ifx∈a⊔b, then a,b⊑x. Therefore, μa,μb≤μx; thus, μx≥μa∨μb.Proposition 8.
LetM;⊓,⊔ be a multilattice. An ℒ-fuzzy subset μ of M is an ℒ-fuzzy m-filter if and only if, for all α∈L, the level set μα is either empty or m-filter.Proof.
⇒) Suppose that μ is an ℒ-fuzzy m-filter. Let α∈L such that μα≠∅. We have to prove that μα is an m-filter.(M1)
Leta,b∈μα and x∈a⊓b. We have μx=μa∧μb≥α because μa≥α and μb≥α. Therefore, x∈μα. Thus, a⊓b⊆μα.(M2)
Leta∈μα, x∈M, and y∈a⊔x. We have μy≥μa≥α, because y⊒a; then, y∈μα. Thus, a⊔x⊆μα.(M3)Leta,b∈M. Suppose that a⊔b∩μα≠∅. There exists x∈a⊔b such that x∈μα; that is, μx≥α. Let y∈a⊔b. We have μx=μy because μ is an ℒ-fuzzy m-filter; then, μy≥α; that is, y∈μα. Thus, a⊔b⊆μα.⇐) Suppose that, for all α∈L, μα=∅ or μα is an m-filter. Let us show that μ is an ℒ-fuzzy m-filter.(FM1)
Letx,y∈M such that x⊑y; that is, x⊔y=y. We have x∈μμx; then, μμx is an m-filter; therefore, x⊔y⊆μμx; that is, y∈μμx. Thus, μx≤μy.(FM2)
Leta,b∈M and x∈a⊓b. We have μa,μb≥μa∧μb; then, a,b∈μα, where α=μa∧μb. Therefore, a⊓b⊆μα; thus, μx≥α=μa∧μb. In addition, x⊑a,b; then, by (FM1), μx≤μa,μb; therefore, μx≤μa∧μb. Thus, μx=μa∧μb.(FM3)
Letx,y∈a⊔b. We have x∈μμx and y∈μμy; then, μμx and μμy are an m-filter. We have x∈a⊔b∩μμx and y∈a⊔b∩μμy; that is, a⊔b∩μμx≠∅ and a⊔b∩μμy≠∅. Then, a⊔b⊆μμx,μμy; therefore, y∈μμx and x∈μμy; that is, μy≥μx and μx≥μy. Thus, μx=μy.Definition 19.
Anℒ-fuzzy subset μ of M is said to be an ℒ-fuzzy filter of ℳ if it is an ℒ-fuzzy deductive system of ℳ, and for all t∈a⊔b⟶b,z∈a⟶a⊓b, μt=μa⟶b=μz, for all a,b∈M.Example 4.
Letℒ be a complete lattice whose Hasse diagram is given in Figure 3. Let ℳ be the residuated multilattice defined in Example 1. The map θ:M⟶L defined by(5)θx=0,ifx∈A∪⊥,β,ifx=b0,γ,ifx=b1,δ,ifx=b2,1,ifx=⊤,is an ℒ-fuzzy filter of ℳ. We have γ and δ are incomparable; then, Imθ is not a chain.
The mapρ:M⟶L defined by(6)ρx=0,ifx∈⊥,a0,α,ifx∈Aa0,β,ifx=b0,γ,ifx=b1,δ,ifx=b2,1,ifx=⊤,is an ℒ-fuzzy p-filter of M which is not an ℒ-fuzzy filter because ρa2⟶a0=ρa0≠ρa4=ρa2⟶a3 and a0∈a2⊓a3.Remark 2.
(i)
If we replace the complete latticeℒ by the lattice 0,1;min,max,0,1, then this definition of ℒ-fuzzy filter of residuated multilattice will coincide with the one given by L. N. Maffeu Nzoda et al. [10].(ii)
Theℒ-fuzzy filter θ defined in Example 5 is an ℒ-fuzzy m-filter.(iii)
Theℒ-fuzzy p-filter ρ defined in Example 5 is not an ℒ-fuzzy m-filter because ρa2∧ρa3≠ρa0 and a0∈a2⊓a3.
The following result characterizes theℒ-fuzzy filter of residuated multilattice in terms of its level subset.Proposition 9.
Letμ be an ℒ-fuzzy subset of ℳ. μ is an ℒ-fuzzy filter of ℳ if and only if, for each α∈L, μα≠∅ implies that it is a filter of ℳ.Proof.
⇒) Suppose that μ is an ℒ-fuzzy filter of ℳ. Let α∈L such that μα≠∅. By Proposition 6, μα is a deductive system because μ is an ℒ-fuzzy deductive system. Let a,b∈M such that a⟶b∈μα. Let t∈a⊔b⟶b and z∈a⟶a⊓b. We have μt=μa⟶b≥α and μz=μa⟶b≥α. Therefore, t∈μα and z∈μα; thus, a⊔b⟶b⊆μα and a⟶a⊓b⊆μα. Hence, μα is a filter of ℳ.⇐) Suppose that, for all α∈L, μα is either empty or a filter of ℳ. By Proposition 6, μ is an ℒ-fuzzy deductive system of ℳ. Let a,b∈M, t∈a⊔b⟶b, and z∈a⟶a⊓b. We have a⟶b∈maxa⊔b⟶band maxa⟶a⊓b. Then, μa⟶b≥μt and μa⟶b≥μz, because μ is an ℒ-fuzzy deductive system of ℳ.
Letα=μa⟶b. We have a⟶b∈μα. Then, a⊔b⟶b and a⟶a⊓b⊆μα, because μα is a filter by hypothesis. Thus, μt≥α and μz≥α; that is, μt≥μa⟶b and μz≥μa⟶b.
Hence,μt=μa⟶b and μz=μa⟶b. Thus, μ is an ℒ-fuzzy filter.Proposition 10.
Leth:M⟶M′ be a homomorphism between residuated multilattices and μ a ℒ-fuzzy filter of M′. Then, the inverse image of μ denoted by h−1μ is a ℒ-fuzzy filter of ℳ, where for all x∈M, h−1μx=μhx.Proof.
Leta,b∈M such that a⊑b. We have ha⊓b⊆ha⊓hb; that is, ha∈ha⊓hb; then, ha⊑hb. Therefore, μha≤μhb because μ is an ℒ-fuzzy filter. Thus, h−1μa≤h−1μb.
Leta,b∈M. We have ha⊙b=ha⊙hb; then, h−1μa⊙b=μha⊙hb=μha∧μhb. Thus, h−1μa⊙b=h−1μa∧h−1μb.
Leta,b∈M, t∈a⊔b⟶b, and z∈a⟶a⊓b. We have h−1μa⟶b=μha⟶b=μha⟶hb. Therefore, h−1μa⟶b=μy and y∈ha⊔hb⟶hb∪ha⟶ha⊓hb, because μ is an ℒ-fuzzy filter. Since h is a residuated multilattices homomorphism, we have ha⊔hb⟶hb⊇ha⊔b⟶b and ha⟶ha⊓hb⊇ha⟶a⊓b. Therefore, h−1μa⟶b=μht and h−1μa⟶b=μhz; that is, h−1μt=h−1μa⟶b=h−1μz.
Thus,h−1μ is an ℒ-fuzzy filter of ℳ.
Anℒ-fuzzy filter of ℳ is also an ℒ-fuzzy m-filter of ℳ, but the converse is not always verified. Indeed, the ℒ-fuzzy m-filter of Example 3 is not an ℒ-fuzzy p-filter of the residuated multilattice of Example 1 because νa4⊙a4=νa1≠νa4. Therefore, it is not an ℒ-fuzzy filter. The following theorem gives the link between the ℒ-fuzzy p-filter, ℒ-fuzzy m-filter, and ℒ-fuzzy filter of a residuated multilattice.Theorem 3.
Letμ be an ℒ-fuzzy subset of ℳ. μ is an ℒ-fuzzy filter of ℳ if and only if(i)
μ is an ℒ-fuzzy deductive system of ℳ;(ii)
μ is an ℒ-fuzzy m-filter of ℳ;(iii)
for alla,b∈M, μx⟶y=μy⟶x, for all x,y∈a∗b, where ∗∈⊔,⊓.Proof.
⇒) Suppose that μ is an ℒ-fuzzy filter of ℳ.(i)
By definition ofℒ-fuzzy filter, μ is an ℒ-fuzzy deductive system of ℳ.(ii)
Let us show thatμ is an ℒ-fuzzy m-filter of ℳ.(FM1)
Letx,y∈M. If x⊑y, then μx≤μy because μ is a deductive system.(FM2)
Letx∈a⊓b. Given that μ is an ℒ-fuzzy deductive system, we have μx≥μa∧μa⟶x=μa∧μa⟶b≥μa∧μb because a⟶b⊒b. By (FM1), μx≤μa∧μb, because x⊑a,b. Hence, μx=μa∧μb.(FM3)
Letx,y∈a⊔b. If a⊔b is a singleton, then x=y and μx=μy. Suppose that a⊔b is not a singleton and x≠y. By Proposition 2, there exists a′,b′∈x⊓y such that x,y∈a′⊔b′. Since μ is an ℒ-fuzzy deductive system, we have μy≥μx∧μx⟶y≥μx∧μx⟶b′=μx∧μa′⟶b′ because y⊒b′⇒x⟶y⊒x⟶b′ and x∈a′⊔b′. From y⊒a′, we obtain μa′⟶b′≥μy⟶b′=μy⟶x≥μx because b′∈x⊓y and y⟶x⊒x. Therefore, μy≥μx. Similarly, we prove that μx≥μy. Thus, μx=μy.Hence,μ is an ℒ-fuzzy m-filter of M.(iii)
Letx,y∈a⊔b. Let us prove that μy⟶x=μx⟶y. If a⊔b is a singleton, then x=y and μx⟶y=μy⟶x. Suppose that x≠y. Then, by Proposition 2 (i), there exists a′,b′∈x⊓y such that x,y∈a′⊔b′. Since x⊒a′, we have μy⟶x≥μy⟶a′=μb′⟶a′≥μx⟶a′ because y∈a′⊔b′ and b′⊑x. We have μx⟶a′=μx⟶y because a′∈x⊓y, and then μy⟶x≥μx⟶y. Similarly, we prove that μx⟶y≥μy⟶x.
Letx,y∈a⊓b. The proof of μy⟶x=μx⟶y is similar to the above by using Proposition 2 (ii).
⇐) Suppose that i, ii, and iii are satisfied. Since μ is an ℒ-fuzzy deductive system, let a,b∈M. We have to prove that, for all t∈a⊔b⟶b and z∈a⟶a⊓b, μt=μa⟶b=μz.
By Proposition3, we have a⟶b∈a⊔b⟶b∩a⟶a⊓b; then, there exist x1∈a⊔b and y1∈a⊓b such that a⟶b=x1⟶b and a⟶b=a⟶y1.
Lett∈a⊔b⟶b. If t=a⟶b, then μt=μa⟶b. If not, t≠a⟶b; that is, there exists x2∈a⊔b, such that t=x2⟶b and x1≠x2. Since x1,x2∈a⊔b, we have μx1⟶x2=μx2⟶x1 by the hypothesis and μx2⟶b≥μx1⟶b∧μx1⟶b⟶x2⟶b because μ is an ℒ-fuzzy deductive system. By property P7 of the pocrim, we have x1⟶b⟶x2⟶b⊒x2⟶x1, and then μt=μx2⟶b≥μx1⟶b∧μx2⟶x1=μx1⟶b∧μx1⟶x2. Since x2⊒b, we have μx1⟶x2≥μx1⟶b. Therefore, μt≥μx1⟶b=μa⟶b. We have a⟶b=maxa⊔b⟶b; then, μa⟶b≥μt. Hence, μa⟶b=μt.
Similarly, we prove that, for allz∈a⟶a⊓b, μa⟶b=μz.
Sinceμ is an ℒ-fuzzy deductive system by hypothesis, we conclude that μ is an ℒ-fuzzy filter.Definition 20.
Anℒ-fuzzy deductive system μ of ℳ is said to be consistent if, for all a,b,c∈M, the following conditions hold:(i)
μx⟶c≥μa⟶c∧μb⟶c, for all x∈a⊔b.(ii)
μc⟶y≥μc⟶a∧μc⟶b, for all y∈a⊓b.Example 5.
Letℒ be a complete lattice whose Hasse diagram is given in Figure 3. Let ℳ be the residuated multilattice defined in Example 1. The only consistent ℒ-fuzzy deductive systems of this residuated multilattice are the constant maps of ℳ to ℒ.Proposition 11.
Every consistentℒ-fuzzy deductive system of ℳ is an ℒ-fuzzy filter of ℳ.Proof.
Letμ be a consistent ℒ-fuzzy deductive system of ℳ. Let a,b∈M, x∈a⊔b, and y∈a⊓b. We have μx⟶b≥μa⟶b∧μb⟶b and μa⟶y≥μa⟶a∧μa⟶b, and then μx⟶b≥μa⟶b and μa⟶y≥μa⟶b because μa⟶a=μb⟶b=μ⊤≥μa⟶b. We have a⟶b=maxa⟶a⊓b=maxa⊔b⟶b, and then μa⟶b≥μx⟶b, μa⟶y. Therefore, μx⟶b=μa⟶b=μa⟶y. Thus, μ is an ℒ-fuzzy filter of ℳ.
The converse of this proposition is not always true, and the following example is an illustration.Example 6.
Theℒ-fuzzy filter θ of Example 5 is not consistent because θa3⟶a2=θa4<θ⊤=θa0⟶a2∧θa1⟶a2, where a3∈a0⊔a1.
In the rest of this paper, we will use the followingℒ-fuzzy subset of M. Let F be a nonempty subset of M and α,β∈L such that α>β. Define the map μFα,β:M⟶L as follows: for all x∈M,(7)μFα,βx=α,ifx∈F,β,ortherwise.Remark 3.
It is easy to see that, for all nonempty subsetF of M, the ℒ-fuzzy subset μFα,β of M is a proper ℒ-fuzzy filter of ℳ if and only if F is a proper filter of ℳ.Proposition 12.
Letμ and η be two ℒ-fuzzy filters of ℳ. Then, the ℒ-fuzzy subset μ∧η of M defined by μ∧ηx=μx∧ηx, for all x∈M, is an ℒ-fuzzy filter of ℳ.Proof.
Letθ=μ∧η. Let x,y∈M such that x⊑y. We have μx≤μy and ηx≤ηy, and then μx∧ηx≤μy∧ηy; that is, θx≤θy.
Letx,y∈M. We have(8)θx⊙y=μx⊙y∧ηx⊙y=μx∧μy∧ηx∧ηy=μx∧ηx∧μy∧ηy=θx∧θy.
Hence,θ is an ℒ-fuzzy deductive system.
Leta,b∈M, t∈a⊔b⟶b, and z∈a⟶a⊓b. We have θt=μt∧ηt and θz=μz∧ηz. Then, θt=μa⟶b∧ηa⟶b=θa⟶b and θz=μa⟶b∧ηa⟶b=θa⟶b, because μ and θ are ℒ-fuzzy filters of ℳ.
Thus,θ=μ∧η is an ℒ-fuzzy filter of ℳ.
In general, givenμ and η two ℒ-fuzzy filters of ℳ, the ℒ-fuzzy subset μ∨η defined by μ∨ηx=μx∨ηx, for all x∈M, is not always an ℒ-fuzzy filter as shown in the following example.Example 7.
Letℒ be a complete lattice whose Hasse diagram is given in Figure 3. Let ℳ be a residuated multilattice defined in Example 1. The map η:M⟶L, defined by(9)ηx=α,ifx∈A∪⊥,β,ifx∈b0,b1,γ,ifx=b2,1,ifx=⊤,is an ℒ-fuzzy filter of ℳ. Consider the ℒ-fuzzy filter θ of ℳ defined in Example 5. We have(10)θ∨ηx=α,ifx∈A∪⊥,β,ifx=b0,γ,ifx=b1,1,ifx∈⊤,b2.
Then,λ=θ∨η is not an ℒ-fuzzy filter of this residuated multilattice because λb1⊙b2=λb0=β≠γ=λb1∧λb2.
LetFFℳ denote the set of all ℒ-fuzzy filters of ℳ. On this set, we define the order relation ≺¯ as follows: for all μ,η∈FFℳ, μ≺¯η if and only if μx≤ηx, for all x∈M.Definition 21.
Letμ:M⟶L be an ℒ-fuzzy subset of ℳ. An ℒ-fuzzy filter η of ℳ is said to be generated by μ if μ≺¯η, and for any ℒ-fuzzy filter θ of ℳ if μ≺¯θ, then η≺¯θ; that is, η is the smallest ℒ-fuzzy filter of ℳ that contains μ. The ℒ-fuzzy filter generated by μ will be denoted by μ^.
For allμ,η∈FFℳ, we define. Let α∈L and Cα be an ℒ-fuzzy subset of M defined by Cαx=α. It is easy to show that FFℳ;∧,∪,C0,C1 is a bounded lattice.


## 4.ℒ-Fuzzy Prime Filter of Residuated Multilattice

This section is devoted to the notions ofℒ-fuzzy prime filter and prime ℒ-fuzzy filter of a residuated multilattice. Their characterizations will be given as well as the relation which links them.Definition 22.
Letμ be a proper ℒ-fuzzy filter of ℳ. μ is an ℒ-fuzzy prime filter if, for all α∈L, μα is a prime filter of ℳ, whenever M≠μα≠∅.Example 8.
Let us consider the residuated multilatticeℳ defined in Example 1 and the lattice ℒ=0,1,min,max,0,1. The map μ:M⟶0,1 defined by(11)μx=0,ifx∈A∪⊥,0,5,ifx∈B∪⊤,is an ℒ-fuzzy prime filter of ℳ.Remark 4.
LetF be a nonempty subset of ℳ. The ℒ-fuzzy subset μFα,β of ℳ is an ℒ-fuzzy prime filter of ℳ if and only if F is a prime filter of ℳ.Proposition 13.
Letμ be a proper ℒ-fuzzy filter of ℳ. If μ is an ℒ-fuzzy prime filter, then Imμ is a chain.Proof.
Letμ be an ℒ-fuzzy prime filter of ℳ. Let x,y∈M and α=μx∨μy. If μα=M, then μx∨μy=μ⊥; that is, μx=μy=μ⊥.
Suppose thatμα≠M. Because ⊤∈μα, we have μα≠∅. Hence, μα is a prime filter. Let a∈x⊔y, and we have a≥x,y; then, μa≥μx,μy; therefore, μa≥α; that is, a∈μα. Thus, x⊔y⊆μα. Since μα is a prime filter, then x∈μα or y∈μα; that is, μx≥μx∨μy≥μy or μy≥μx∨μy≥μx. Therefore, μx≥μy or μy≥μx; hence, Imμ is a chain.
The following result characterizes anℒ-fuzzy prime filter of residuated multilattice.Theorem 4.
Letμ be a proper ℒ-fuzzy filter of ℳ. μ is an ℒ-fuzzy prime filter of ℳ if and only if Imμ is a chain and ∧t∈x⊔yμt=μx∨μy for all x,y∈M.Proof.
Suppose thatμ is an ℒ-fuzzy prime filter of ℳ. By Proposition 13, Imμ is a chain. Let x,y∈M. By hypothesis, μ is a ℒ-fuzzy m-filter; then, for all t∈x⊔y, μt≥μx∨μy; thus, ∧t∈x⊔yμt≥μx∨μy. In addition, let α=∧t∈x⊔yμt. We have μa≥α, for all a∈x⊔y. Then, x⊔y⊆μα. If μα=M, then x∈μα and y∈μα. If not, μα is a prime filter of ℳ; therefore, x∈μα or y∈μα. In any case, we have μx≥α or μy≥α. Because μx∨μy≥μx,μy, we have μx∨μy≥∧t∈x⊔yμt. Hence, ∧t∈x⊔yμt=μx∨μy.
Conversely, suppose thatImμ is a chain, and for all x,y∈M, ∧t∈x⊔yμt=μx∨μy. Let α∈L such that M≠μα≠∅. Let x,y∈M such that x⊔y⊆μα and x∉μα. We have μt≥α for all t∈x⊔y, and then μx∨μy=∧t∈x⊔yμt≥α. Since Imμ is a chain and μx≱α, we have μy=μx∨μy≥α; that is, y∈μα. Thus, μα is a prime filter.
Now we introduce the notion of primeℒ-fuzzy filter of residuated multilattice ℳ, which is a prime element in the lattice FFℳ of ℒ-fuzzy filters of ℳ.Definition 23.
Letμ be a proper ℒ-fuzzy filter of ℳ. μ is a prime ℒ-fuzzy filter of ℳ if, for all θ,η∈FFℳ, θ∧η≺¯μ implies θ≺¯μ or η≺¯μ.Example 9.
Let us consider the residuated multilatticeℳ defined in Example 1 and the lattice ℒ=0,1;min,max,0,1. The map μ:M⟶0,1 defined by(12)μx=1,ifx∈B∪⊤,0,otherwise,is a prime ℒ-fuzzy filter of ℳ.Lemma 3.
Letμ be a proper ℒ-fuzzy filter of ℳ. If μ is a prime ℒ-fuzzy filter of ℳ, then the following conditions hold:(1)
μ⊤=1.(2)
Imμ=μ⊥,1.(3)
μ⊥ is an ∧-prime.Proof.
(1)
Suppose that1≠μ⊤. Let θ and η be two ℒ-fuzzy subsets of M defined by(13)θx=1,ifx∈μμ⊤,0,ifx∉μμ⊤,andηx=μ⊤ for all x∈M. Then, θ is an ℒ-fuzzy filter by Proposition 8 and Proposition 9. η is an ℒ-fuzzy filter because it is a constant map.We have(14)θ∧ηx=μ⊤,ifx∈μμ⊤,0,ifx∉μμ⊤.For allx∈M, 0≤μx. Therefore, θ∧ηx≤μx, for all x∉μμ⊤. If x∈μμ⊤, then μ⊤≤μx; that is, θ∧ηx≤μx, for all x∈μμ⊤. Hence, θ∧η≺¯μ. Since μ is a proper, there exists a∈M, such that μ⊤>μa. Hence, μa=μ⊤≰μa; that is, η⋠μ. In addition, θ⊤=1≰μ⊤, because 1≠μ⊤; that is, θ⋠μ. Therefore, θ∧η≺¯μ, θ⋠μ, and η⋠μ; this contradicts the fact that μ is a prime ℒ-fuzzy filter of ℳ. Hence, μ⊤=1.(2)
Leta∈M∖⊥ such that μa≠μ⊤. Suppose that μa≠μ⊥; that is, μ⊥<μa. Let θ and η be two ℒ-fuzzy subsets of M defined by(15)θx=1,ifx∈μμa,μ⊥,ifx∉μμa,andηx=μa, for all x∈M. Then, θ is an ℒ-fuzzy filter by Remark 3 and 2.4; η is an ℒ-fuzzy filter because it is a constant map.We have(16)θ∧ηx=μa,ifx∈μμa,μ⊥,ifx∉μμa.It is easy to show thatθ∧η≺¯μ. Since μ⊥<μa<μ⊤=1, we have θ⋠μ and η⋠μ; this contradicts the fact that μ is a prime ℒ-fuzzy filter of ℳ. Thus, μa=μ⊥ for all a∈M∖⊥; hence, Imμ=μ⊥,1.(3)
Letα,β∈L such that α∧β≤μ⊥. Let θ and η be two ℒ-fuzzy subsets of M defined by θx=α and ηx=β, for all x∈M. We have θ∧ηx=α∧β≤μ⊥≤μx for all x∈M; then, θ∧η≺¯μ. Since θ and η are ℒ-fuzzy filters of ℳ and μ is a prime ℒ-fuzzy filter, we have θ≺¯μ or η≺¯μ. Therefore, θ⊥=α≤μ⊥ or η⊥=β≤μ⊥; hence, μ⊥ is an ∧-prime.
It is clear that anℒ-fuzzy prime filter is not always a prime ℒ-fuzzy filter because the example of ℒ-fuzzy prime filter defined in Example 8 is not a prime ℒ-fuzzy filter. In addition, a prime ℒ-fuzzy filter is not always an ℒ-fuzzy prime filter according to Example 9. The following theorem gives the relation existing between ℒ-fuzzy prime filter and prime ℒ-fuzzy filter of residuated multilattice.Remark 5.
By Proposition13 and Remark 4, a prime ℒ-fuzzy filter μ of ℳ is an ℒ-fuzzy prime filter of ℳ if and only if μ1 is a prime filter of ℳ.Lemma 4.
LetF be a filter of ℳ, α be an ∧-prime element of L, and μ be an ℒ-fuzzy subset of M defined by(17)μx=1,ifx∈F,α,otherwise.
Then,μ is a prime ℒ-fuzzy filter of ℳ.Proof.
We have1>α, and F is a filter, then by Remark 3, and μ is an ℒ-fuzzy filter of ℳ. Let θ and η be two ℒ-fuzzy filters of ℳ such that θ∧η≺¯μ. We have θ∧ηx≤μx, and then θx∧ηx≤μx.
Ifx∈F, then θx≤μx and ηx≤μx because μx=1.
Else,θx∧ηx≤α, and then θx≤α or ηx≤α because α is an ∧-prime element of L. In any case, we have θ≺¯μ or η≺¯μ. Thus, μ is a prime ℒ-fuzzy filter of ℳ.
The following result characterizes primeℒ-fuzzy filter of ℳ.Theorem 5.
Letμ be a proper ℒ-fuzzy filter of ℳ. μ is a prime ℒ-fuzzy filter of ℳ if and only if Imμ=μ⊥,1 and μ⊥ is an ∧-prime element of L.Proof.
Letμ be a proper ℒ-fuzzy filter of ℳ.
Suppose thatμ is a prime ℒ-fuzzy filter of ℳ, and then by Lemma 3, Imμ=μ⊥,1 and μ⊥ is an ∧-prime element of L.
Conversely, suppose thatImμ=μ⊥,1 and μ⊥ is an ∧-prime element of L. We have(18)μx=1,ifx∈μ1,μ⊥,ifx∉μ1.
By Proposition9 and Lemma 4, μ is a prime ℒ-fuzzy filter of ℳ.


## 5.ℒ-Fuzzy Maximal Filter of Residuated Multilattices

Definition 24.
LetF be a proper filter of ℳ. F is a maximal filter of ℳ if, for all filter G of ℳ, F⊆G implies that G=F or G=M.Definition 25.
A properℒ-fuzzy filter μ of ℳ is called ℒ-fuzzy maximal filter of ℳ if for each α∈L, when μα is nontrivial, then μα is a maximal filter of ℳ.Example 10.
LetPE;∩,∪,θ,E be a complete lattice, where E=α,β. Let ℳ be a residuated multilattice defined in Example 1. Consider the map μ:M⟶PE defined by, for all x∈M,(19)μx=α,ifx∈M∖⊤,E,ifx=⊤.
μ is a PE-fuzzy maximal filter of ℳ.Theorem 6.
Letμ be a proper ℒ-fuzzy filter of ℳ. μ is an ℒ-fuzzy maximal filter if and only if, for all x,y∈M, μx≰μy implies x=⊤ or μμx is a maximal filter of ℳ.Proof.
Suppose thatμ is an ℒ-fuzzy maximal filter. Let x,y∈M such that μx≰μy. Let α=μx and β=μy. If α and β are incomparable, then μα and μβ are nontrivial filters. By hypothesis, μα and μβ are maximal filters. Therefore, μμx is a maximal filter. Else, if μy<μx, then μμx⊈μμy⊆M; that is, ∅≠μμx≠M. Suppose that x≠⊤; then, μμx≠⊤; that is, μμx is a nontrivial filter. By hypothesis, μμx is a maximal filter of ℳ.
Conversely, let us assume that, for allx,y∈M, μx≰μy implies x=⊤ or μμx is a maximal filter of ℳ. Let α∈L such that μα is nontrivial; that is, ⊤≠μα≠M. There exists x,y∈M such that x≠⊤, x∈μα, and y∉μα; that is, x≠⊤, μx≥α, and μy≱α. Then, there exists x,y∈M such that x≠⊤ and μx≰μy. Hence, by hypothesis, μμx is a maximal filter of ℳ. Moreover, μμx⊆μα because μx≥α. Since μα≠M and μμx is maximal, we have μμx=μα. Thus, μα is a maximal filter of ℳ.Proposition 14.
Letμ be an ℒ-fuzzy maximal filter of ℳ. In the ordered set Imμ,≤, every element of Imμ∖μ⊥,μ⊤ is both an atom and a coatom.Proof.
Letα∈Imμ∖μ⊥,μ⊤.
Let us show thatα is an atom. Suppose that there exists β∈Imμ such that μ⊥<β<α<μ⊤. We have μμ⊤⊈μα⊈μβ⊈μμ⊥=M. Moreover, μx=α≰β=μy, and then by Theorem 6 and since α≠μ⊤, μα is a maximal filter. We have μα⊈μβ and μβ≠M; this contradicts the fact that μα is a maximal filter. Then, there is no β∈Imμ such that μ⊥<β<α. Thus, α is an atom of Imμ,≤.
Similarly, we prove thatα is a coatom of Imμ,≤.
Now we will define the maximal element ofFFℳ.Definition 26.
Letμ be a proper ℒ-fuzzy filter of ℳ. μ is a maximal ℒ-fuzzy filter of ℳ if for every ℒ-fuzzy filter η of ℳ, μ≺¯η implies that η=C1 or η=μ.Example 11.
LetPE;∩,∪,∅,E be a complete lattice, where E=α,β. Let ℳ be a residuated multilattice defined in Example 1. Consider the map θ:M⟶PE defined by(20)θx=α,ifx∈A,E,otherwise.
θ is a maximal PE-fuzzy filter of ℳ.Remark 6.
In Example10, the PE-fuzzy maximal filter μ of ℳ is not a maximal PE-fuzzy filter because we have μ≺¯θ, θ≠CE, and θ≠μ, where θ is a PE-fuzzy filter of ℳ defined in Example 11.
In the rest of this section, we assume thatℒ is a completely meet-distributive lattice.Lemma 5.
Letμ be an ℒ-fuzzy filter of ℳ and α∈L. Then, μ∨Cα is an ℒ-fuzzy filter of ℳ.Proof.
Letη=μ∨Cα. Let x,y∈M such that x⊑y. We have μx≤μy, and then ηx=μx∨α≤μy∨α=ηy.
Letx,y∈M. We have μx⊙y∨α=μx∧μy∨α=μx∨α∧μy∨α because ℒ is a meet-distributive. Then, ηx⊙y=ηx∧ηy.
Leta,b∈M, t∈a⊔b⟶b, and z∈a⟶a⊓b. We have ηa⟶b=μa⟶b∨α; then, ηa⟶b=μt∨α=ηt and ηa⟶b=μz∨α=ηz. Thus, η is an ℒ-fuzzy filter of ℳ.Proposition 15.
Letμ be a maximal ℒ-fuzzy filter of ℳ. Then, Imμ=μ⊥,1 and μ⊥ is a coatom of L.Proof.
Sinceμ is proper, Imμ≥2. Let α∈Imμ such that α≠μ⊥; that is, there exists a∈M such that α=μa≠μ⊥. By Lemma 5, η=μ∨Cα is an ℒ-fuzzy filter, and μ≺¯η. We have η⊥=α≠μ⊥; then, η≠μ. Therefore, η=C1 because μ is maximal. Hence, 1=C1⊥=η⊥=μ⊥∨Cα⊥=μ⊥∨μa=μa≤μ⊤; that is, α=1=μ⊤. Therefore, Imμ=μ⊥,1.
Now let us show thatμ⊥=β is a coatom. Suppose that β is not a coatom; then, there exists γ∈L such that β<γ<1. Consider the map ν:M⟶L defined by, for all x∈M,(21)νx=1,ifμx=1,γ,ifμx=β.
ν is an ℒ-fuzzy filter, because ν=μF1,γ where F=μ1. We have μ≺¯ν, μ≠ν, and ν≠C1, then contradicting the maximality of μ. Thus, β=μ⊥ is a coatom in L.Proposition 16.
Ifℒ is a distributive lattice, then every maximal ℒ-fuzzy filter of ℳ is a prime ℒ-fuzzy filter of ℳ.Proof.
Letℒ be a distributive lattice. Let μ be a maximal ℒ-fuzzy filter of ℳ. By Proposition 15, Imμ=μ⊥,1, where μ⊥ is a coatom of L. By Theorem 5, we only have to prove that μ⊥ is an ∧-prime element of L. Let α,β∈L such that α∧β≤μ⊥. Suppose that α≰μ⊥ and β≰μ⊥; that is, μ⊥<α∨μ⊥ and μ⊥<β∨μ⊥. Then, α∨μ⊥=1 and β∨μ⊥=1, because μ⊥ is a coatom. Therefore, μ⊥=α∧β∨μ⊥=α∨μ⊥∧β∨μ⊥=1, which contradicts the fact that μ⊥ is a coatom. Hence, μ⊥ is an∧-prime element of L. Thus, μ is a prime ℒ-fuzzy filter of ℳ.
Since in a lattice an∧-prime element is not always a coatom, a prime ℒ-fuzzy filter is not always a maximal ℒ-fuzzy filter.Lemma 6.
Ifμ is a maximal ℒ-fuzzy filter of ℳ, then μ1 is a maximal filter of ℳ.Proof.
Letμ be a maximal ℒ-fuzzy filter of ℳ. Then, by Proposition 15, Imμ=μ⊥,1 and μ⊥ is a coatom. Suppose that μ1 is not a maximal filter of ℳ; then, there exists a proper filter H of ℳ such that μ1⊈H. Let η be the map μH1,μ⊥. Let x∈M. If x∉H, then x∉μ1; that is, μx≤≠1. Therefore, μx=μ⊥ and μx≤ηx=μ⊥. Else, if x ∊ H, then ηx=1 and μx≤ηx. Hence, μ≺¯η.
On the other hand, we haveμ1⊈Hμ1⊄H, and then μ≠η. And since H ≠ M, we have η≠C1. Hence, μ≺¯η, μ≠η, and η≠C1. This contradicts the fact that μ is a maximal ℒ-fuzzy filter of ℳ. Thus, μ1 is a maximal filter of ℳ.Remark 7.
By Definition25, it is clear that if ℒ is a completely meet-distributive lattice, then a maximal ℒ-fuzzy filter of a residuated multilattice ℳ is an ℒ-fuzzy maximal filter of ℳ.Theorem 7.
A properℒ-fuzzy filter μ of ℳ is a maximal ℒ-fuzzy filter if and only if μ=ηG⊥,α, where G is a maximal filter of ℳ and α is coatom in L.Proof.
Suppose thatμ is a maximal ℒ-fuzzy filter of ℳ. By Proposition 15 and Lemma 6, Imμ=1,μ⊥. α=μ⊥, is a coatom, and G = μ1 is a maximal filter. We have μ(x) = 1 or μ(x) = α. If μ(x) = 1, then x∈G and ηG⊥,α=1. Else, if μ(x) = α, then x∉G and ηG⊥,α=α. Thus, μ=ηG⊥,α.
Suppose thatμ=ηG⊥,α, where G is a maximal filter of ℳ and α is coatom in L. We have(22)μx=1,ifx∈G,α,otherwise.Then, G = μ1.
Letθ be a proper ℒ-fuzzy filter of ℳ such that μ≺¯θ. If x∈G, then μ(x) = 1 and μx≤θx; that is, μx=θx. Else, if x∉G, then μx=α≤θx. Since α is a coatom, then θx=α or θx=1; if θx=1, then x∈θ1 and x∉G; that is, G=μ1⊈θ1, hence contradicting the maximality of G. Therefore, θx=α=μx. Then, θx=α=μx; thus, μ is a maximal ℒ-fuzzy filter of ℳ.


## 6.ℒ-Fuzzy Cosets of an ℒ-Fuzzy Filter

Definition 27.
Letℳ be a residuated multilattice. Let μ be an ℒ-fuzzy filter of ℳ and a∈M. A ℒ-fuzzy coset of the ℒ-fuzzy filter μ is the ℒ-fuzzy subset μa of ℳ defined by μax=μa⟶x∧μx⟶a, for all x∈M.
Letμ be an ℒ-fuzzy filter of ℳ and a∈L. Consider the relationa∼ defined on ℳ by, for all a,b∈M,a∼bα if and only if μab≥α.Remark 8.
Letμ be an ℒ-fuzzy filter of ℳ and α∈L such that α≤μ⊤. Then,∼a is an equivalence relation on ℳ.Proposition 17.
Letμ be an ℒ-fuzzy filter of ℳ and α∈L, such that α≤μ⊤. Then,∼α is an equivalence relation on ℳ compatible with ⊙ and ⟶.Proof.
By Remark8,∼α is an equivalence relation on ℳ.
Letx,y,z∈Msuch that xa∼y. By P6 and P7, we have x⊙z⟶y⊙z,x⟶z⟶y⟶z,z⟶x⟶z⟶y⊒x⟶y. Then, μx⊙z⟶y⊙z,μx⟶z⟶y⟶z,μz⟶x⟶z⟶y≥μx⟶y. Similarly, we have μy⊙z⟶x⊙z,μy⟶z⟶x⟶z,μz⟶y⟶z⟶x. Therefore, μx⊙zy⊙z,μx⟶zy⟶z,μz⟶yz⟶y≥μxy≥α. Thus, x⊙z∼ay⊙z,x⟶z∼ay⟶z, and z⟶x∼αz⟶y.Theorem 8.
Letμ be an ℒ-fuzzy filter of ℳ; then, ∼μ⊤ is a congruence relation on ℳ.Proof.
Let us show, by using Theorem2, that ∼μ⊤ is a congruence of multilattice M;⊓,⊔. By Proposition 17, ∼μ⊤ is an equivalence relation on M; then, (i) and (iii) of Theorem 2 are verified.
(ii) Let a,b∈M. Suppose that a∼μ⊤b; then, μab=μ⊤. For all x∈a⊓b and y∈a⊔b, we have x⊑y; then, μx⟶y=μ⊤. Since μ is an ℒ-fuzzy filter, μy⟶b=μa⟶b and μb⟶x=μb⟶a, for all x∈a⊓b and y∈a⊔b. By P5, we have for all x∈a⊓b and y∈a⊔b, μy⟶x≥μy⟶b⊙b⊙x≥μy⟶b∧μb⟶x; then, μy⟶x≥μa⟶b∧μb⟶a=μ⊤. Thus, x∼μ⊤, for all x∈a⊓b and y∈a⊔b.
Conversely, assume that there existx∈a⊓b and y∈a⊔b with x∼μ⊤y. By hypothesis, μx⟶y=μy⟶x=μ⊤. We have b⊒x and y⊒a; then, a⟶b⊒a⟶x⊒y⟶x; that is, μa⟶b≥μy⟶x=μ⊤; that is, μa⟶b=μ⊤. Similarly, we prove that μb⟶a=μ⊤. Therefore, a∼μ⊤b.
(iv) Let a,b∈M such that a⊒b and a∼μ⊤b. Since μx≤μ⊤, for all x∉M, and by definition of ∼μ⊤, we have μa⟶b=μb⟶a=μ⊤.(i)
Let us prove thata⊓c∼^μ⊤b⊓c, for all c∈M.Letx∈a⊓c. We have x⊑a⊑b and x⊑c; then, there exists y∈b⊓c such that x⊑y; then, μx⟶y=μ⊤. We have y⊑b; then, y⟶a⊒b⟶a and μy⟶a=μ⊤. Since μ is an ℒ-fuzzy filter, we have μy⟶x=μy⟶a=μ⊤, since x∈a⊓y by Lemma 1. Hence, x∼μ⊤y. Thus, for all x∈a⊓c, there exists y∈b⊓c such that x∼μ⊤y.Lety∈b⊓c. Check x∈a⊓c such that x∼μ⊤y. For all z∈a⊓c, we have z≤b,c; then, z≤y or z and y are incomparable; that is, z≤y or z∈b⊓c.Suppose that there existsz∈a⊓c such that z≤y. Then, by Lemma 1, we have z∈a⊓y. Therefore, μy⟶z=μy⟶a≥μb⟶a=μ⊤ because y≤b and a∼μ⊤b; that is, μy⟶z=μ⊤. Moreover, we haveμz⟶y=μ⊤; then, z∼μ⊤y. In this case, take x = z.Suppose that, for allz∈a⊓c, z∈b⊓c; that is, a⊓c⊆b⊓c. If y∈a⊓c take x = y because ∼μ⊤ is reflexive. If not, a and y are incomparable. Let t∈a⊓y; we have t≤a,y≤a,c; then there exists ω∈a⊓c such that t≤ω. By Lemma 1, t∈y⊓ω. We have μy⟶ω=μt⟶t=μy⟶a and μy⟶a≥μb⟶a=μ⊤; that is, μy⟶ω=μ⊤. Since a⊓c⊆b⊓c, we have μy⟶ω=μω⟶y because y,ω∈b⊓c and μ is an ℒ-fuzzy filter of ℳ. Therefore, μy⟶ω=μ⊤=μω⟶y; that is, ω∼μ⊤y. In this case, take x=ω.(ii)
Similarly, we prove thata⊔c∼^μ⊤b⊔c, for all c∈M.
Hence,∼μ⊤ is a congruence on multilattice M;⊓,⊔. And by Proposition 17, ∼μ⊤ is a congruence of pocrim M. Thus, ∼μ⊤ is a congruence of residuated multilattice ℳ.Lemma 7.
Letμ be an ℒ-fuzzy filter of ℳ and a,b∈M. μa=μb if and only if a∼μ⊤b.Proof.
Leta,b∈M. If μa=μb, then μab=μbb=μ⊤; thus, a∼μ⊤b.
Conversely, ifa∼μ⊤b, then μab=μ⊤; that is, μa⟶b=μb⟶a=μ⊤. Since μ is an ℒ-fuzzy filter, for all x∈M, μx⟶a≥μx⟶b∧μx⟶b⟶x⟶a≥μx⟶b∧μb⟶a, because x⟶b⟶x⟶a⊒b⟶a. Therefore, μx⟶a≥μx⟶b. Similarly, we prove that μx⟶b≥μx⟶a. Thus,μx⟶b=μx⟶a. We also have μb⟶x=μa⟶x; then for all x∈M, μax=μbx.Remark 9.
Letμ be an ℒ-fuzzy filter of ℳ, and let M/μ denote the set of all coset of μ; that is, M/μ=μa,a∈M.
For allx,y,a,b∈M, if μx= μ and μy= μb, then μx⊙y=μa⊙b, μx⟶y=μa⟶b, μx⊓y=μa⊓b, and μx⊔y=μa⊔b, where μA=μa,a∈A, for all A⊆M. Therefore, ⊙,⟶,⊓,and⊔ are well defined on M/μ.
Thus, for anyμa,μb∈M/μ, we define μa⊙μb=μa⊙b; μa⟶μb=μa⟶b; μa⊓μb=μa⊓b; and μa⊔μb=μa⊔b.Proposition 18.
Letμ be an ℒ-fuzzy filter of ℳ. For all a,b∈M, μa=μa⊓μb if and only if μb=μa⊔μb.Proof.
Leta,b∈M. Suppose that μa=μa⊓μb; we haveμa=μx,x∈a⊓b. Then, for all x∈a⊓b, μx=μa; that is, a∼μ⊤x. Let y∈a⊔b; we have b⊑y; then, μb⟶y=μ⊤. Let x∈a⊓b. Since μ is an ℒ-fuzzy filter, we have μy⟶b=μa⟶b=μa⟶x=μ⊤, because a∼μ⊤x. We have thus proved that, for all y∈a⊔b, b∼μ⊤y; that is, μy=μb. It follows that μa⊔μb=μy,y∈a⊔b=μb.
Similarly, we prove that ifμb=μa⊔μb, then μa=μa⊓μb.
Letμ be an ℒ-fuzzy filter of ℳ. Let a,b∈M. We have μa≤μb if and only if μax≤μbx, for all x∈M. Particularly, if μa≤μb, then μ⊤=μaa≤μba; that is, a∼μ⊤b. It follows that μa≤μb if and only if μa=μb. Therefore, the classical ordering relation ≤ on (M/μ) leads to equality. Consequently, we need to build a suitable order relation on (M/μ).Proposition 19.
Letμ be an ℒ-fuzzy filter of ℒ. The relation defined for all a,b∈M, μa≺¯μb if and only if μa=μa⊓μb is an ordering relation on (M/μ).Proof.
≤ is easily seen to be reflexive and antisymmetric.
Leta,b,c∈M such that μa≺¯μb and μb≺¯μc. Then, for all y∈a⊓b, z∈b⊓c, a∼μ⊤y, and b∼μ⊤z. Let us show that μa≺¯μc; that is, μa=μa⊓c. Let x∈a⊓c. We have μx⟶a=μ⊤ and μa⟶x=μa⟶c; then,μa⟶≥μa⟶b∧μa⟶b⟶a⟶c≥μa⟶b∧μb⟶c. For all y∈a⊓b, μa⟶b=μa⟶y=μ⊤, because a∼μ⊤y. Also, we have for all z∈b⊓c, μb⟶c=μb⟶z=μ⊤, because b∼μ⊤z. Then, μa⟶x=μ⊤; that is, a∼μ⊤x. Hence, μa=μx, for all x∈a⊓c; that is, μa⊓μc=μx,x∈a⊓c=μa. Therefore, μa≺¯μc.
Thus,≺¯ is an ordering relation on (M/μ).
Now, we have all the required properties to prove that the set of allℒ-fuzzy cosets of an ℒ-fuzzy filter of a residuated multilattice is a residuated multilattice. First, we prove that it is a multilattice.Proposition 20.
Letμ be an ℒ-fuzzy filter of ℳ. M/μ;≺¯,⊓,⊔,μ⊤ is a multilattice.Proof.
Leta,b,x∈M such that μa≺¯μx and μb≺¯μx. Check α∈μa⊔μb such that α≺¯μx. Let y∈a⊓x and z∈b⊓x. We have a∼μ⊤y and b∼μ⊤z by hypothesis. Since y⊑x and z⊑x, then there exists ω∈y⊔z such that ω≤x. Since a∼μ⊤y and b∼μ⊤z, then a⊔b∼^μ⊤b⊔y and b⊔y∼^μ⊤y⊔z. Therefore, a⊔b∼^μ⊤y⊔z. We have ω∈y⊔z; then, there exists t∈a⊔b such that t∼μ⊤ω; that is, μt⟶ω=μ⊤=μω⟶t. We have ω⊑x; then μt⟶ω≤μt⟶x. Thus, μt⟶x=μ⊤.
Letc∈t⊓x; we have μt⟶c=μt⟶x=μ⊤. Also, c⊑t; then, μc⟶t=μ⊤. Therefore, for all c∈t⊓x, c∼μ⊤t; that is, μc=μt. Thus, μt⊓μx=μc,c∈t⊓x=μt; that is, μt≺¯μx. Take α=μt.
Similarly, we prove the dual.We now prove that the multilattice of allℒ-fuzzy cosets of an ℒ-fuzzy filter of a residuated multilattice is a residuated multilattice.Proposition 21.
Letμ be an ℒ-fuzzy filter of ℳ. Then, M/μ;≺¯,⊙,⟶,⊓,⊔,μ⊤ is a residuated multilattice.Proof.
We have thatM/μ;≺¯,⊓,⊔,μ⊤ is an ordered multilattice by Proposition 20 and M/μ;⊙,μ⊤ is a commutative monoid with neutral element μ⊤ by definition of ⊙.
Letμa,μb,μc∈M/μ. Suppose that μa⊙μc≺¯μb; that is, μx=μa⊙c, for all x∈a⊙c⊓b. Hence, for all x∈a⊙c⊓b, μa⊙c⟶b=μa⊙c⟶x=μ⊤ because a⊙c∼μ⊤x and μ is an ℒ-fuzzy filter of ℳ. Let y∈c⊓a⟶b; we have μc⟶y=μc⟶a⟶b=μa⊙c⟶b=μ⊤, by property P4 of pocrim. Since y⊑c, then μy⟶c=μ⊤. Therefore, c∼μ⊤y; that is, μy=μc for all y∈c⊓a⟶b. Thus, μc⊓μa⟶μb=μy,y∈c⊓a⟶b=μc; that is, μc≺¯μa⟶μb.
Suppose thatμc≺¯μa⟶μb; then, for all x∈c⊓a⟶b, μc⟶a⟶b=μc⟶x=μ⊤ because x∼μ⊤c. Let y∈a⊙c⟶b. We have μa⊙c⟶y=μa⊙c⟶b=μc⟶a⟶b=μ⊤. Since y≤a⊙c, we have μy⟶a⊙b=μ⊤. Hence, y∼μ⊤a⊙c. Thus, μa⊙μc⊓μb=μy,y∈a⊙c⊓b=μa⊙c; that is, μa⊙μc≤μb.Define the mapμ^:M⟶M/μ by μ^a=μa, for all a∈M.Theorem 9.
Letμ be an ℒ-fuzzy filter of ℳ. Then,(i)
μ^ is a surjective homomorphism of residuated multilattice;(ii)
Kerμ^=μμ⊤;(iii)
M/μ≅M/μμ⊤, where M/μμ⊤ is the set of all equivalence classes of the congruence relation ∼μ⊤.Proof.
Letμ be an ℒ-fuzzy filter of ℳ.(i)
By Lemma 5.5,M/μ is a residuated multilattice. For all a,b∈M, we have μ^a⊙b=μa⊙b=μa⊙μb=μ^a⊙μ^b; μ^a⟶b=μa⟶b=μ^a⟶μ^b=μ^a⟶μ^b; μ^a⊓b=μa⊓b=μa⊓¯μb=μ^a⊓¯μ^b; μ^a⊔b=μa⊔b=μa⊔¯μb=μ^a⊔¯μ^b; andμ^⊤=μ⊤. Thus, by definition of μ^, it is a surjective homomorphism of residuated multilattice.(ii)
x∈Kerμ^ if and only if μ^x=μ⊤ iff μx=μ⊤ iff x∼μ⊤⊤ iff μx⊤=μ⊤ iff μx=μ⊤ iff x∈μμ⊤. Thus, Kerμ^=μμ⊤.(iii)
Let us define the mapφ:M:μ⟶M/μμ⊤ by φμa=a=x∈M,x∼μ⊤a. We have μa=μb iff a∼μ⊤b iff a=b. Then, φ is well defined, and it is injective. φ is surjective by definition. Thus, M/μ≅M/μμ⊤.


## 7. Conclusion

We have introduced the notion ofℒ-fuzzy filter of residuated multilattice, which generalizes the work of Maffeu et al. [12] by replacing the closed unit interval [0, 1] by a complete lattice ℒ. Besides that, we have studied the notion of ℒ-fuzzy prime filter of residuated multilattice and prime ℒ-fuzzy filter. It appears that a prime ℒ-fuzzy filter μ is an ℒ-fuzzy prime if the 1-cut set of μ is a prime filter. The concept of maximal ℒ-fuzzy filter and ℒ-fuzzy maximal filter was defined; it follows that a maximal ℒ-fuzzy filter is also an ℒ-fuzzy maximal filter if ℒ is distributive, but the converse is not always verified. Finally, we introduce the notion of ℒ-fuzzy cosets of an ℒ-fuzzy filter, and we prove that the set of all ℒ-fuzzy cosets of an ℒ-fuzzy filter is also a residuated multilattice. For future work, we will study the notions of ℒ-fuzzy congruence and ℒ-fuzzy homomorphism of residuated multilattice.



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