# L-Fuzzy Congruences and L-Fuzzy Kernel Ideals in Ockham Algebras

Authors: Teferi Getachew Alemayehu; Derso Abeje Engidaw; Gezahagne Mulat Addis
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebraA,f, whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.


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## Body

## 1. Introduction

Zadeh [1] introduced the concept of fuzzy sets, which has been found to be very useful in diversely applied areas of science and technology. In the last two decades, several articles have been written on the application of fuzzy sets. For instance, in medical diagnosis, Kaur and Chaira [2] proposed a novel fuzzy clustering approach that enhances the quality of vague CT scan/MRI image before segmentation. In addition, many authors (e.g., [3–6]) applied the theory of fuzzy sets in decision-making. Furthermore, the theory of fuzzy sets has been conveniently and successfully applied in abstract algebra. The study of fuzzy subalgebras of various algebraic structures has been started after Rosenfeld wrote his seminal paper [7] on fuzzy subgroups. His paper has provided sufficient motivation to researchers to study fuzzy subalgebras of different algebraic structures. For instance, fuzzy ideals and fuzzy filters of MS-algebras [8–10], some generalizations of fuzzy ideals in distributive lattices [11–13], fuzzy ideals and fuzzy filters of partially ordered sets [14, 15], and fuzzy ideals of universal algebras [16–18] are some of recent works on fuzzy subalgebraic structures.As an extension of Zadeh’s fuzzy set theory [1], Atanassov [19] introduced the intuitionistic fuzzy sets (IFS), characterized by a membership function and a nonmembership function. Further investigation has been made by other scholars to apply the theory of intuitionistic fuzzy sets in the class of BG-algebras [20], B-algebras [21], and BCK-algebras [22] as well.A fuzzy congruence relation on general algebraic structures is a fuzzy equivalence relations which is compatible (in a fuzzy sense) with all fundamental operations of the algebra. The notions of fuzzy congruence relations were studied in various algebraic structures: in semigroups (see [23, 24]), in groups, rings, and semirings (see [25–30]), in modules and vector spaces (see [31, 32]), in lattices (see [33, 34]), in almost distributive lattices and MS-algebras (see [35, 36]), and, more generally, in universal algebras (see [37–39]).The notion of Ockham algebras was initially introduced by Berman [40] in 1977. In simple terminology, an Ockham algebra is a bounded distributive lattice equipped with a dual endomorphism. Blyth and Silva [41] have studied and characterized kernel ideals in Ockham algebra. The purpose of this paper is to apply the theory of L-fuzzy sets in the class of Ockham algebras, where L is a complete lattice satisfying the infinite meet distributive law:(1)a∧supS=supa∧x:x∈S,for any a∈L and S⊆L. To be specific, we study L-fuzzy congruences and L-fuzzy kernel ideals of Ockham algebras and investigate their properties. We also derive some equivalent conditions for every L-fuzzy ideal of an Ockham algebra A to become an L-fuzzy kernel ideal. We give an internal characterization for the smallest and the largest L-fuzzy congruences on A having a given L-fuzzy ideal as a kernel.


## 2. Preliminaries

This section contains some basic definitions and results which will be used the sequel.Definition 1 (see [42]).
An Ockham algebra is an algebraA;∧,∨,f,0,1 of type 2,2,1,0,0 in which A;∧,∨,0,1 is a bounded distributive lattice and x↦fx is a unary operation on A such that f0=1,f1=0, and the following de Morgan laws hold(2)∀x,y∈Afx∨y=fx∧fy,fx∧y=fx∨fy.For simplicity, we denote an Ockham algebraA;∧,∨,f,0,1 by a pair A,f.Definition 2 (see [42]).
A congruence relation on an Ockham algebraA,f is a lattice congruence ϑ on A such that x,y∈A,x,y∈ϑ⇒fx,fy∈ϑ.Definition 3 (see [41]).
By an ideal of an Ockham algebraA,f, we mean an ideal of A as a distributive lattice. Moreover, an ideal I of an Ockham algebra A is called a kernel ideal if there exists a congruence ϑ on A,f such that(3)I=ϑ0=x∈A:x,0∈ϑ.By anL-fuzzy subset μ of a nonempty X, we mean a mapping from X into L. The set of all L-fuzzy subsets of X is denoted by LX.Definition 4 (see [43]).
Letμ,σ in LX. Then, the Cartesian product of μ and σ, denoted by μ×σ, is defined by, for all x,y∈X,(4)μ×σx,y=μx∧μy.The union and intersection of any familyμii∈Δ of L-fuzzy subsets of X, respectively, denoted by ∪i∈Δμi and ∩i∈Δμi, are defined by(5)∪i∈Δμix=supi∈Δμix,∩i∈Δμix=infi∈Δμix,for all x∈X, respectively.Definition 5 (see [44]).
For anyμ and σ in LX, define a binary relation “⊆” on LX by(6)μ⊆σif and only ifμx≤σx,for allx∈X.It can be easily verified that⊆ is a partial order on the set LX of L-fuzzy subsets of X and the poset LX,⊆ forms a complete lattice, in which, for any μii∈Δ⊆LX,(7)supi∈Δμi=∪i∈Δμi,infi∈Δμi=∩i∈Δμi.The partial ordering “⊆” is called the pointwise ordering.Forμ∈LX and α∈L, the set,(8)μα=x∈X:μx≥α,is called the α-level subset of μ, and for each x∈X, we have(9)μx=∨α∈L:x∈μα.For anyα∈L, we write α¯ to denote the constant L-fuzzy subset of X which maps every element of X onto α.Definition 6 (see [7]).
Letf be a function from X into Y, and let μ be an L-fuzzy subset of X. Then, the image of μ under f, denoted by fμ, is an L-fuzzy subset of Y given by, for all y∈Y,(10)fμy=supμx:x∈f−1y,iff−1y≠∅,0,otherwise.
The preimage ofσ under f, symbolized by f−1σ, is an L-fuzzy subset of X and(11)f−1σx=σfxfor allx∈X.Definition 7 (see [33]).
AnL-fuzzy subset μ of a lattice X with 0 is said to be an L-fuzzy ideal of X if μ0=1 and μa∨b=μa∧μb, for all a,b∈X.
Dually, anL-fuzzy subset μ of a lattice X with 1 is said to be an L-fuzzy filter of X if μ1=1 and μa∧b=μa∧μb, for all a,b∈X.AnL-fuzzy ideal (respectively, filter) μ of X is said to be proper if it is not a constant map 1¯. By an L-fuzzy binary relation on a nonempty set X, we mean an L-fuzzy subset of X×X. For an L-fuzzy binary relation Θ on X and each α∈L, the set,(12)Θα=x,y∈X×X:Θx,y≥α,is called the α-level binary relation of Θ on X.Definition 8 (see [45]).
AnL-fuzzy relation Θ on a nonempty set X is said to be(1)
Reflexive ifΘx,x=1, for all x∈X(2)
Symmetric ifΘx,y=Θy,x, for all x,y∈X(3)
Transitive if, for eachx,y∈X, Θx,y≥Θx,z∧Θz,y, for all z∈X
A reflexive, symmetric, and transitiveL-fuzzy relation on X is called an L-fuzzy equivalence relation on X.


## 3.L-Fuzzy Congruences of Ockham Algebras

In this section, we give various characterizations of anL-fuzzy congruence relation of an Ockham algebra. Throughout this section and the rest, A stands for an Ockham algebra A,f.Definition 9.
AnL-fuzzy equivalence relation Θ on A is called an L-fuzzy congruence relation on A if it satisfies the following conditions:(1)
Θa∧c,b∧d∧Θa∨c,b∨d≥Θa,b∧Θc,d(2)
Θfa,fb≥Θa,b, for all a,b,c,d∈AAnL-fuzzy equivalence relation on A that satisfies condition (1) is called a lattice L-fuzzy congruence on A. The following two lemmas give important characterization for L-fuzzy congruence relations in Ockham algebras.Lemma 1.
AnL-fuzzy relation Θ on A is an L-fuzzy congruence relation on A if and only if Θα is a congruence relation on A, for all α∈L.Lemma 2.
AnL-fuzzy equivalence relation Θ on A is an L-fuzzy congruence relation on A if and only if, for any a,b,x∈A,(13)Θfa,fb∧Θa∧x,b∧x∧Θa∨x,b∨x≥Θa,b.For anya∈L and L-fuzzy congruence relation Θ, define an L-fuzzy subset Θa of A by(14)Θax=Θa,x,for allx∈A.We callΘa an L-fuzzy congruence class of Θ determined by a, and in particular, Θ0 is called the kernel of Θ. One can easily observe that kernel Θ0 of Θ is an L-fuzzy ideal of A.Lemma 3.
LetΘ be a fuzzy congruence on A. For any x,y∈A, the following holds(1)
Θx=Θy if and only if Θx,y=1(2)
EitherΘx=Θy or there exists α∈L−1 such that Θx∩Θy⊆α¯Let us putA/Θ=Θx:x∈A and define binary operations ∧,∨ and a unary operation f on A/Θ by(15)Θx∧Θy=Θx∧y,Θx∨Θy=Θx∨y,fΘx=Θfx.It is routine to verify thatA/Θ,∧,∨,f,Θ0,Θ1 is an Ockham algebra, and it is called the quotient Ockham algebra of A modulo Θ. For an L-fuzzy subset μ of A, we write ΘLμ (respectively, ΘlatLμ) to denote the smallest L-fuzzy congruence (respectively lattice L-fuzzy congruence) on A containing μ×μ. It was proved in [34] that, for any x,y∈A,(16)ΘlatLμx,y=∨μi:x∨i=y∨i,whenever μ is an L-fuzzy ideal of A and(17)ΘlatLμx,y=∨μj:x∧j=y∧j,whenever μ is an L-fuzzy filter of A.For a givenL-fuzzy ideal μ of A, we shall now investigate the smallest L-fuzzy congruence ΘLμ of A containing μ×μ.Definition 10.
AnL-fuzzy subset μ of A is called an L-fuzzy down set (respectively, L-fuzzy up set) if, for any x,y∈A, μx≥μy (respectively, μx≤μy) whenever x≤y.Lemma 4.
Letμ be an L-fuzzy subset of A. Then, the L-fuzzy subset μ↓ of A defined by(18)μ↓x=∨μy:x≤y,for allx∈A,is the smallest L-fuzzy down set containing μ.Proof.
Letx∈A. Since x≤x, we clearly have(19)μ↓x=∨μy:x≤y≥μx.
Thus,μ⊆μ↓. Let x,y∈A such that x≤y. Now,(20)μ↓x=∨μz:x≤z≥∨μz:y≤z=μ↓y.
Therefore,μ↓ is an L-fuzzy down set of A. Let σ be any L-fuzzy down set of A such that μ⊆σ. For any x∈A, we have(21)μ↓x=∨μy:x≤y≤∨σy:x≤y=σx.
This implies thatμ↓⊆σ. Therefore, μ↓ is the smallest L-fuzzy down set of A containing μ.Dually, we have the following lemma.Lemma 5.
Letμ be an L-fuzzy subset of A. Then, the L-fuzzy subset μ↑ of A defined by(22)μ↑x=∨μy:y≤x,for allx∈A,is the smallest L-fuzzy up set containing μ.Lemma 6.
Letμ be an L-fuzzy ideal of A. For each nonnegative integer n, define μ2n=f2nμ↓,μ2n+1=f2n+1μ↑. Then, μ2n is an L-fuzzy ideal and μ2n+1 is an L-fuzzy filter of A.Proof.
Now, since(23)μ2n0=f2nμ↓0=∨f2nμy:0≤y≥f2nμ0Since0≤0=∨μa:a∈f2n−10Sincef2n−10≠∅≥μ0=1Since0∈f2n−10.
We haveμ2n0=1.
Again, letx,y∈A. Then, if ∀wx≤w, f2n−1w=∅ or ∀zy≤z, f2n−1z=∅, then we have(24)μ2nx∧μ2ny=f2nμ↓x∧f2nμ↓y=∨f2nμw:x≤w∧∨f2nμz:y≤z=0≤μ2nx∨y.
If∃wx≤w, f2n−1w≠∅ and ∃zy≤z, f2n−1z≠∅, we have(25)μ2nx∧μ2ny=f2nμ↓x∧f2nμ↓y=∨f2nμw:x≤w∧supf2nμz:y≤z=∨μa:x≤f2na,a∈A∧∨μb:y≤f2nb,b∈A=∨μa∧μb:x≤f2na,y≤f2nba,b∈A≤∨μa∨b:x∨y≤f2na∨b,a,b∈A≤f2nμ↓x∨y=μ2nx∨y.
Again, asμ2n is an L-fuzzy down set, we clearly have μ2nx∧μ2ny≥μ2nx∨y, and hence, μ2nx∨y=μ2nx∧μ2ny. Therefore, μ2n is an L-fuzzy ideal of A. Analogously, we can prove that μ2n+1 is an L-fuzzy filter of A.Lemma 7.
Letμ be any L-fuzzy ideal of A. Put μ∘∘=∨n≥0μ2n and μ∘=∨n≥0μ2n+1. Then,(26)μ∘fx≥μ∘∘x,μ∘fx≥μ∘x,for allx∈A.Proof.
Letx∈A. If ∀yx≤y, f2n−1y=∅, we have(27)μ∘x=∨n≥0μ2nx=∨n≥0∧n≥0μ2nx=∨n≥0∧n≥0f2nμ↓x=∨n≥0∧n≥0∨f2nμy:x≤y=0≤μ∘fx.
If∃yx≤y, f2n−1y≠∅, then(28)μ∘x=∨n≥0μ2nx=∨n≥0∧n≥0μ2nx=∨n≥0∧n≥0f2nμ↓x=∨n≥0∧n≥0∨f2nμy:x≤y=∨n≥0∧n≥0∨∨μz:f2nz=y:x≤y≤∨n≥0∧n≥0∨∨μz:f2n+1z=fy:fy≤fx=∨n≥0∧n≥0∨f2n+1μfy:fy≤fx≤∨n≥0∧n≥0f2n+1μ↑fx=∨n≥0∧n≥0μ2n+1fx=∨n≥0μ2n+1fx=μ∘fx.
Similarly, we can show thatμ∘∘fx≥μ∘x.Lemma 8.
Letμ be an L-fuzzy ideal of A. Then, the L-fuzzy binary relation ΩL on A defined by(29)ΩLμx,y=∨μ∘∘a∧μ∘b:x∨a∧b=y∨a∧b,for all x,y∈A is an L-fuzzy congruence relation on A.Proof.
Clearly,ΩLμ is reflexive and symmetric. We show that it is transitive. Let x,y,z∈A. If x∨a∧b=y∨a∧b,y∨c∧d=z∨c∧d, then it can be easily verified that x∨a∨c∧b∧d=z∨a∨c∧b∧d. Thus, we have(30)ΩLμx,y∧ΩLμy,z=∨μ∘∘a∧μ∘b:x∨a∧b=y∨a∧b∧∨μ∘∘c∧μ∘d:y∨c∧d=z∨c∧d=∨μ∘∘a∧μ∘∘c∧μ∘b∧μ∘d:x∨a∧b=y∨a∧b,y∨c∧d=z∨c∧d≤∨μ∘∘a∨c∧μ∘b∧d:x∨a∨c∧b∧d=z∨a∨c∧b∧d=ΩLμx,z.
Therefore,ΩLμ is transitive, and hence, it is an L- fuzzy equivalence relation on A.
Next, we show thatΩLμ satisfies the substitution properties. If x∨a∧b=y∨a∧b,z∨c∧d=w∨c∧d, then, after routine work, we obtain(31)x∧z∨a∨c∧b∧d=y∧w∨a∨c∧b∧d,ΩLμx,y∧ΩLμz,w=∨μ∘a∧μ∘b:x∨a∧b=y∨a∧b∧∨μ∘∘c∧μ∘d:z∨c∧d=w∨c∧d=∨μ∘∘a∧μ∘∘c∧μ∘b∧μ∘d:x∨a∧b=y∨a∧b,z∨c∧d=w∨c∧d≤∨μ∘∘a∨c∧μ∘b∧d:x∧z∨a∨c∧b∧d=y∧w∨a∨c∧b∧d=ΩLμx∧z,y∧w.
Similarly, we can show that(32)ΩLμx,y∧ΩLμz,w≤ΩLμx∨z,y∨w.
Finally, ifx∨a∧b=y∨a∧b, then it can be easily verified(33)fx∨fb∧fa∨fb=fy∨fb∧fa∨fb.
Now,(34)ΩLμx,y=∨μ∘∘a∧μ∘b:x∨a∧b=y∨a∧b≤∨μ∘∘fb∧μ∘fa:fx∨fb∧fa∨fb=fy∨fb∧fa∨fb≤∨μ∘∘fb∧μ∘fa∨fb:fx∨fb∧fa∨fb=fy∨fb∧fa∨fb,a,b∈A=ΩLμfx,fy.
Thus,ΩLμ is an L-fuzzy congruence on A.Theorem 1.
For anyL-fuzzy ideal μ of A,(35)ΩLμ=ΘlatLμ∘∘∨ΘlatLμ∘,where the join is taken in the lattice of L-fuzzy congruences on A.Proof.
We first show thatΘlatLμ∘∘⊆ΩLμ and ΘlatLμ∘⊆ΩLμ. Now, for any x,y∈A, we have(36)ΘlatLμ∘∘x,y=∨μ∘∘s:x∨s=y∨s≤∨μ∘∘s∧μ∘fs:x∨s∧fs=y∨s∧fs≤ΩLμx,y.
Hence,ΘlatLμ∘∘⊆ΩLμ. Similarly, we have ΘlatLμ∘⊆ΩLμ. This implies that(37)ΘlatLμ∘∘∨ΘlatLμ∘⊆ΩLμ.
To establish the reverse inclusion, letx,y∈A. Then,(38)ΩLμx,y=∨μ∘∘s∧μ∘t:x∨s∧t=y∨s∧t≤∨μ∘∘s∧μ∘t:x∧x∨s∧t=x∧y∨s∧t≤∨μ∘t:x∧t=x∧y∨s∧t=ΘlatLμ∘x,x∧y∨s,ΩLμx,y=∨μ∘∘s∧μ∘t:x∨s∧t=y∨s∧t≤∨μ∘∘s∧μ∘t:x∧x∨s∧t=x∧y∨s∧t=∨μ∘∘s∧μ∘t:x∧t=x∧y∨s∧t≤∨μ∘∘s∧μ∘t:x∧t∨s=x∧y∨s∧t∨s≤∨μ∘∘s:x∧y∨s∨s=x∧y∨s=ΘlatLμ∘∘x∧y∨s,x∧y.
This implies that(39)ΩLμx,y≤ΘlatLμ∘x,x∧y∨s∧ΘlatLμ∘∘x∧y∨s,x∧y≤ΘlatLμ∘∨ΘlatLμ∘∘x,x∧y.
Similarly,ΩLμx,y≤ΘlatLμ∘∨ΘlatLμ∘∘x∧y,y. Again,(40)ΩLμx,y≤ΘlatLμ∘∨ΘlatLμ∘∘x,x∧y∨ΘlatLμ∘∨ΘlatLμ°°x∧y,y≤ΘlatLμ∘∨ΘlatLμ∘∘x,y.
Hence,ΩLμ=ΘlatLμ∘∨ΘlatLμ∘∘.Corollary 1.
ΩLμ is the smallest L-fuzzy congruence on A containing μ×μ.


## 4. L-Fuzzy Kernel Ideals

In this section, we studyL-fuzzy kernel ideals in Ockham algebras and give several characterizations for them.Definition 11.
AnL-fuzzy ideal μ on A is called a kernel L-fuzzy ideal if μ=Θ0 for some L-fuzzy congruence Θ of A.The following observation is immediate.Lemma 9.
Ifμ is an L-fuzzy ideal of A such that f2μ⊆μ, then, for every n,(41)f2nμ⊆μ,f2n+1μ⊆fμ↑,from which it follows that(42)μ∘∘=μ,μ∘=fμ↑.The following lemma gives an internal characterization forL-fuzzy kernel ideals in Ockaham algebras.Lemma 10.
AnL-fuzzy ideal μ of A is an L-fuzzy kernel ideal if and only if it satisfies the following properties:(1)
f2μ⊆μ(2)
fμ↑t∧μx∧t≤μx, for all x,t∈AProof.
Letμ be an L-fuzzy kernel ideal of A. Then, there exists an L-fuzzy congruence Θ on A such that μ=Θ0. Let x∈A. If f2−1x=∅, then f2μx=0≤μx. Let f2x−1≠∅, and consider the following:(43)f2μx=∨μa:x=f2a=∨Θa,0:x=f2a≤Θf2a,0=Θx,0μx.
Hence,f2μx⊆μx, for all x∈A. Therefore, (1) holds. To prove (2), let x,t∈A. If f−1t=∅, clearly (2) holds. Suppose that f−1t≠∅. Then,(44)fμ↑t∧μx∧t=∨fμa:a≤t∧μx∧t=∨∨μy:fy=a:a≤t∧μx∧t=∨μy:fy≤t∧μx∧t=∨Θy,0:fy≤t∧Θt∧x,0≤Θfy,1∧Θt∧x,0≤Θt,1∧Θt∧x,0≤Θt∧x,x∧Θt∧x,0≤Θx,0=μx.
Conversely, suppose thatμ is an L-fuzzy ideal of A satisfying conditions (1) and (2). Consider the L-fuzzy congruence ΩLμ given in Lemma 8. Now, we claim that μ=ΩLμ0. Let x∈A. Then,(45)μx=μx∧μ0=μ×μx,0⊆ΩLμx,0=ΩLμ0x,and hence, μ⊆ΩLμ0. To show the other inclusion,(46)ΩLμ0x=ΩLμx,0=∨μ∘∘s∧μ∘t:x∨s∧t=0∨s∧t=∨μs∧fμ↑t:x∨s∧t=s∧tby Lemma9≤∨μs∧t∧fμ↑t:x∨s∧t=s∧t=∨μx∨s∧t∧fμ↑t:x∨s∧t=s∧t≤μx∨s≤μx.
Hence,ΩLμ0⊆μ. Therefore, ΩLμ0=μ.Corollary 2.
Ifμ is a proper kernel L-fuzzy ideal of A, then(47)μ∩fμ⊆α¯,for some α∈L−1.Proof.
Suppose thatμ is a proper kernel L-fuzzy ideal of A. Then, μ=Θ0, for some L-fuzzy congruence Θ of A and μa≠1, for some a∈A. Put α=μa. Now, we claim that μ∩fμ⊆α¯. Let x∈A. Then, the claim clearly holds if f−1x=∅. Suppose that f−1x≠∅. Then,(48)μ∩fμx=μx∧fμx=μx∧∨μy:x=fy=Θ0x∧∨Θ0y:x=fy=Θx,0∧∨Θy,0:x=fy≤Θx,0∧Θx,1≤Θ1,0=μ1≤μa=α.
Hence, the claim is true.Lemma 11.
Arbitrary intersection of a family of kernelL-fuzzy ideals of A is a kernel L-fuzzy ideal.We denote the set of all kernelL-fuzzy ideals of A by ℱℐKA.Theorem 2.
The setℱℐKA of kernels L-fuzzy ideal of A ordered by set inclusion forms a complete lattice.Consider the subsetℱℐ2A=μ∈ℱℐA:f2μ⊆μ. Then, we have the following results.Theorem 3.
(1)
ℱℐ2A is a complete sublattice of ℱℐA(2)
Ifμ is a kernel L-fuzzy ideal of A, then μ∈ℱℐ2AProof.
Letμi:i∈Δ be a nonempty subset of ℱℐ2A. Then, f2μi⊆μi, for all i∈Δ. Now, we claim that ∩i∈Δμi∈ℱℐ2A.
Letx∈A. Suppose that f2−1x=∅. Then,(49)f2∩i∈Δμix=0≤∩i∈Δμix.
Now, suppose thatf2−1x≠∅. Then,(50)f2∩i∈Δμix=sup∩i∈Δμia:x=f2a=supinfi∈Δμia:x=f2a=infi∈Δsupμia:x=f2a=infi∈Δf2μix≤infi∈Δμix=∩i∈Δμix.
Hence, the claim is true. SinceχA is greatest element of ℱℐ2A, we have ℱℐ2A is a complete sublattice of ℱℐA.
It follows from Lemma10.Lemma 12.
Letμ be an L-fuzzy ideal of A. Then, the L-fuzzy subset μ˜ of A defined by(51)μ˜x=∨μi∧μx∧fi:i∈A,for allx∈A,is an L-fuzzy ideal of A containing μ.Proof.
Suppose thatμ be an L-fuzzy ideal of A. Now, since(52)μ˜0=∨μi∧μ0∧fi:i∈A=∨μi:i∈A≥μ0=1.
We haveμ˜0=1. Let x,y∈A. Then,(53)μ˜x∧μ˜y=∨μi∧μx∧fi:i∈A∧∨μj∧μy∧fj:j∈A=∨μi∧μx∧fi∧μj∧μy∧fj:i,j∈A≤∨μi∨j∧μx∧fi∨j∧μy∧fi∨j:i,j∈A≤∨μi∨j∧μx∨y∧fi∨j:i∨j∈A=μ˜x∨y.
Again since(54)μ˜x=∨μi∧μx∧fi:i∈A≥∨μi∧μx∨y∧fi:i∈A=μ˜x∨y.
Similarly, we haveμ˜y≥μ˜x∨y, and hence, μ˜x∧μ˜y≥μ˜x∨y. Therefore, μ˜x∨y=μ˜x∧μ˜y. Again, since, for any x∈A,(55)μ˜x=∨μi∧μx∧fi:i∈A≥μ0∧μx∧f0=1∧μx∧1=μx,we have μ⊆μ˜. Therefore, μ˜ is an L-fuzzy ideal of A containing μ.In the following theorem, we give a necessary and sufficient condition for anL-fuzzy ideal of an Ockham algebra to be a kernel L-fuzzy ideal.Theorem 4.
AnL-fuzzy ideal μ of A is a kernel L-fuzzy ideal of A if and only if f2μ⊆μ and μ=μ˜.Proof.
Suppose thatμ is a kernel L-fuzzy ideal of A. Then, by Lemmas 10 and 12, we have f2μ⊆μ and μ⊆μ˜. Now, since, for any x∈A,(56)μ˜x=∨μi∧μx∧fi:i∈A≤∨fμfi∧μx∧fi:i∈A≤∨fμ↑fi∧μx∧fi:∈A≤μx,we have μ˜⊆μ. Hence, f2μ⊆μ and μ=μ˜.
Conversely, suppose thatμ is an L-fuzzy ideal of A satisfying the given condition. Then, by Lemma 11, it is enough to show that(57)fμ↑t∧μx∧t≤μx,for allx,t∈A.
Letx,t∈A if f2−1t=∅; clearly, (57) holds. Suppose that f2−1t≠∅. Then,(58)fμ↑t∧μx∧t=∨fμa:a≤t∧μx∧t=∨∨μy:fy=a:a≤t∧μx∧t=∨μy:fy≤t∧μx∧t=∨μy∧μx∧t:fy≤t≤∨μy∧μx∧fy:fy≤t≤μ˜x=μx.
Hence,μ is a kernel L-fuzzy ideal of A.Theorem 5.
Letμ be an L-fuzzy ideal of A. Then,(1)
Ifμ∈ℱℐ2A, then μ˜∈FI2A(2)
The mappingξ:ℱℐ2A⟶ℱℐ2A defined by ξμ=μ˜ is a closure on ℱℐ2L and Imξ=ℱℐKAProof.
Letμ∈ℱℐ2A. Then, f2μ⊆μ. Let x∈A. Then, if f2−1x=∅, obviously, μ˜∈ℱℐ2A. Suppose that f2−1x=∅. Then,(59)f2μ˜x=∨μ˜y:x=f2y=∨∨μi∧μy∧fi:i∈A:x=f2y=∨μi∧μy∧fi:x=f2y,i∈A≤∨f2μf2i∧f2μf2y∧fi:x=f2y,i∈A=∨f2μf2i∧f2μf2y∧ff2i:x=f2y,i∈A=∨μf2i∧μf2y∧ff2i:x=f2y,i∈A=∨μf2i∧μx∧ff2i:f2i∈A≤μ˜x.
Hence,f2μ˜⊆μ˜, i.e., μ˜∈ℱℐ2A.
Sinceμ⊆μ˜=ξμ, ξ is extensive. Also, let μ,ν∈ℱℐ2A such that μ⊆ν. Then,(60)ξμx=μ˜x=∨μi∧μx∧fi,i∈A≤∨νi∧νx∧fi,i∈A=ν˜x=ξμx.
Thus,ξμ⊆ξν, and hence, ξ is isotone. Finally, we show that ξ is idempotent.
Letx∈A. Then,(61)ξξμx=∨ξμs∧ξμx∧fs:s∈A=∨∨μi∧μs∧fi:i∈A∧∨μj∧μx∧fs∧fj:j∈A:s∈A=∨μi∧μs∧fi∧μj∧μx∧fs∧fj:s,i,j∈A≤∨μf2i∧μs∧fi∨j∧μx∧fs∧fj:s,i,j∈A≤∨μx∧fj∧f2i∧μs∧fi∨j∧μx∧fs∧fj:s,i,j∈A≤∨μx∧fj∧f2i∨x∧fs∧fj∧μs∧fi∨j:s,i,j∈A=∨μx∧fs∧fi∨j∧μs∧fi∨j:s,i,j∈A≤ξμx.
Thus,ξξμ⊆ξμ. As ξμ⊆ξξμ, and we have ξξμ=ξμ. Therefore, ξ is idempotent. This proves the result.


## 5. Smallest and LargestL-Fuzzy Congruences with a Given L-Fuzzy Kernel Ideal

In this section, we describe the smallest and the largestL-fuzzy congruences on an Ockham algebra having a given L-fuzzy kernel ideal.From Lemmas8 and 9, we can easily observed that, for any kernel L-fuzzy ideal μ the smallest L-fuzzy congruence on A with kernel μ is ΩLμ which is given by(62)ΩLμx,y=∨μa∧fμ↑b:x∨a∧b=y∨a∧b.We can in fact establish the following simpler version.Theorem 6.
Ifμ∈ℱℐKA, then the smallest L-fuzzy congruence on A with kernel μ is the L-fuzzy relation ΦLμ is given by(63)ΦLμx,y=∨μi:x∨i∧fi=y∨i∧fi,i∈Afor allx∈A.Proof.
To prove it enough to show thatΦLμ=ΩLμ, since μi≤f−1fμi=fμfi, we have(64)ΦLμx,y=∨μi:x∨i∧fi=y∨i∧fi,i∈A=∨μi∧fμfi:x∨i∧fi=y∨i∧fi,i∈A≤ΩLμx,y.Hence,ΦLμ⊆ΩLμ.Sinceμ is kernel L-fuzzy ideal,(65)ΩLμx,y=∨μi∧fμ↑j:x∨i∧j=y∨i∧j,i,j∈A,ΩLμx,y=∨μi∧fμ↑j:x∨i∧j=y∨i∧j,i,j∈A=∨μi∧∨μk:fk≤j:x∨i∧j=y∨i∧j,i,j∈A=∨μi∧μk:x∨i∧fk=y∨i∧fk,i,k∈A≤∨μi∨k:x∨i∨k∧fi∨k=y∨i∨k∧fi∨k,i,k∈A=ΦLμx,y.Hence,ΩLμ=ΦLμ. Thus, ΦLμ is the smallest L-fuzzy congruence with kernel μ.Having described the smallestL-fuzzy congruence on A with a given kernel L-fuzzy ideal, we now proceed to determine the largest such L-fuzzy congruence.For a givenL-fuzzy ideal μ of A, each a∈A, and for each n∈ℕ, consider the L-fuzzy subset of A defined by(66)Wa,nμx=μfna∧xfor allx∈A,and an L-fuzzy binary relation Θnμ on A defined by, for all(67)a,b∈AΘnμa,b=supWa,nμx∧Wb,nμx:x∈A.Lemma 13.
Θnμ is a lattice L-fuzzy congruence of A and ∩n∈ℕΘnμ is an L-fuzzy congruence of A.Proof.
For alla,x∈A,(68)Θnμa,a=supWa,nμx∧Wb,nμx:x∈A=supμfna∧x∧μfnb∧x:x∈A≥μfna∧0∧μfnb∧0=μ0=1.
This impliesΘnμ is reflexive. Clearly, Θnμ is symmetric:(69)Θnμa,b∧Θnμb,c=supWa,nμx∧Wb,nμx:x∈A∧supWb,nμy∧Wc,nμy:y∈A=supWa,nμx∧Wb,nμx∧Wb,nμy∧Wc,nμy:x,y∈A≤supWa,nμx∧Wc,nμy:x,y∈A≤supWa,nμx∧y∧Wc,nμx∧y:x,y∈A=Θnμa,c.
This impliesΘnμ is transitive.(70)Θnμa∨c,b∨d=supWa∨c,nμx∧Wb∨d,nμx:x∈A=supμfna∨c∧x∧μfnb∨d∧x:x∈AIfn−even=supμfna∨fnc∧x∧μfnb∨fnd∧x:x∈A=supμfna∧x∨fnc∧x∧μfnb∧x∨fnd∧x:x∈A≥supμfna∧x∧μfnc∧x∧μfnb∧x∧μfnd∧x:x∈A=supμfna∧x∧μfnb∧x∧supμfnc∧x∧μfnd∧x:x∈A=supWa,nμx∧Wb,nμx::x∈A∧supWc,nμx∧Wd,nμx:x∈A=Θnμa,b∧Θnμc,d.
Also, ifn is odd(71)Θnμa∨c,b∨d=supμfna∧fnc∧x∧μfnb∧fnd∧x:x∈A≥supμfna∧x∧μfnb∧x:x∈A=Θnμa,b.
Similarly,Θnμa∨c,b∨d≥Θnμc,d. This implies Θnμa∨c,b∨d≥Θnμa,b∧Θnμc,d. Hence, Θnμ is compatible with ∨. Similarly, Θnμ is compatible with ∧. Hence, Θnμ is a lattice L-fuzzy congruence of A.
PutΘμ=∩n∈ℕΘnμ. Then, clearly Θμ is a lattice L-fuzzy congruence of A. Next, we show that, for any a,b∈A,(72)Θμfa,fb≥Θμa,b,Θμfa,fb=∩n∈ℕΘnμfa,fb=infn∈ℕΘnμfa,fb=infn∈ℕΘnμfa,fb=infn∈ℕsupμffna∧x∧μffnb:x∈A≥infn∈ℕsupμfna∧x∧μfnb∧x:x∈A=infn∈ℕθnμa,b=θμa,b.
Thus,θμa,b≤θμfa,fb. Hence, θμ is an L-fuzzy congruence of A.In the next theorem, we describe the largestL-fuzzy congruence on A containing a given L-fuzzy ideal μ as its kernel.Theorem 7.
Ifμ∈ℱℐKA, then the largest L-fuzzy congruence on A with kernel μ is Θμ.Proof.
First, we prove thatkerΘμ=μ. For any x∈A,(73)μx≤μfnx≤μfnx∧a=μfnx∧a=Wx,nμa.
SinceW0,nμa=1. This implies(74)μx≤Wx,nμa∧W0,nμa≤infn∈NsupWx,nμa∧W0,nμa:a∈A=infn∈Nθnμx,0=θμx,0=kerθμx.
This impliesμ⊆kerθμ.
Conversely,(75)kerθμx=infn∈NsupWx,nμa∧W0,nμa:a∈A≤supWx,nμa∧W0,nμa:a∈A=supμfnx∧a∧μfn0∧a:a∈A≤μx.
This implieskerθμ⊆μ. Hence, kerθμ=μ.
Finally, we prove thatθμ is the largest L-fuzzy congruence of A. Now, let Φ be any L-fuzzy congruence on A with kernel μ. We show that Φ⊆Θμ. For a,b,x∈A, Φa,0=kerΦa=μa≤μa∧x=Wa,0μx and Φ0,b=kerΦb=μb≤μb∧x=Wb,0μx. This implies Φa,0∧Φ0,b≤Wa,0μx∧Wb,0μx≤supWa,nμx∧Wb,nμx:x∈Afor alln∈W.
It follows that(76)Φa,b≤infn∈WsupWa,nμx∧Wb,nμx:x∈A=θμa,b.
Hence,θμ is the largest L-fuzzy congruence on A having μ as a kernel.


## 6. Conclusion

In this paper, we studyL-fuzzy congruences and L-fuzzy kernel ideals in Ockham algebras and investigate their properties. A set of equivalent conditions is derived for an L-fuzzy ideal μ of A to become an L-fuzzy kernel ideal. We obtain the smallest, respectively, the largest L-fuzzy congruences on A having a given L-fuzzy ideal as a kernel and describe it using algebraic operations in a fuzzy setting.It is under investigation by the authors to characterize those subvarieties of the class of Ockham algebras usingL-fuzzy congruence relations.



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