# Kernel L-Ideals and L-Congruence on a Subclass of Ockham Algebras

Authors: Teferi Getachew Alemayehu; Derso Abeje Engidaw; Gezahagne Mulat Addis
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we studyL-congruences and their kernel in a subclass Kn,0 of the variety of Ockham algebras A. We prove that the class of kernel L-ideals of an Ockham algebra forms a complete Heyting algebra. Moreover, for a given kernel L-ideal ξ on A, we obtain the least and the largest L-congruences on A having ξ as its kernel.


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## Body

## 1. Introduction

The concept of an Ockham algebra was first introduced by Berman [1], in 1977. Next, it has been studied by Urquhart et al. [2], Goldberg et al. [3, 4], and Blyth and Varlet [5]. Blyth and Silva [5] presented the concept of kernel ideals in Ockham algebra. Wang et al. [6] presented Congruences and kernel ideals on a subclass Kn,0 of the variety of Ockham Algebras (in which h2n=idA). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras are some of the well-known subvarieties of Ockham algebra. We see [7] for the basic concepts of the class of Ockham algebras.On the other side, for the first time, the concept of fuzzy sets was presented by Zadeh as an extension of the classical notion of set theory [8]. He defined a fuzzy subset of a nonempty set K as a function from K to 0,1. Goguen in [9] presented the notion of L-fuzzy subsets by replacing the interval 0,1 with a complete lattice L in the definition of fuzzy subsets. Swamy and Swamy [10] studied that complete lattices that fulfill the infinite meet distributive law are the most appropriate candidates to have the truth values of general fuzzy statements.The study of fuzzy subalgebras of different algebraic structures has been begun after Rosenfeld presented his paper [11] on fuzzy subgroups. This paper has provided sufficient motivation to researchers to study the fuzzy subalgebras of different algebraic structures.Fuzzy congruence relations on algebraic structures are fuzzy equivalence relations that are compatible (in a fuzzy sense) with all fundamental operations of the algebra. The concept of fuzzy congruence relations was presented in different algebraic structure: in semigroups (see [12, 13]), in groups, semirings, and rings (see [14–19]), in modules and vector spaces (see [20, 21]), in lattices (see [22, 23]), in universal algebras (see [24, 25]), and more recently in MS-algebras and Ockham Algebras (see [26–28]).Initiated by the above results, we present KernelL-ideals and L-Congruence on a subclass ℱℐkA of Kn,0 of the variety of Ockham algebras and study their characteristics. We prove that the class of kernel L-ideal ℒℐkA of Kn,0-algebra A forms Heyting algebras. Also, we get the least and the biggest L-congruences, respectively, on Kn,0-algebra A having a given L-ideal as an L-kernel.


## 2. Preliminaries

This section contains basic definitions and important results which will be used in the sequel.Definition 1..
(see [5]). An algebra A;∧,∨,h,0,1 of type 2,2,1,0,0 is said to be an Ockham algebra if(1)
A,∧,∨,0,1 is a bounded distributive lattice(2)
h:A↦A is a dual endomorphismIn this paper, for simplicity, any Ockham algebraA;∧,∨,h,0,1 is denoted by a pair A,h. If the dual endomorphism on A satisfies that h2n=idA, then this subclass is the Berman subclass Kn,0.Definition 2.
(see [5]). An equivalence relation Ψ on A is a congruence on A,h if it is lattice congruence on A and for every a,b∈A,(1)a,b∈Ψ⇒ha,hb∈Ψ.Definition 3.
(see [7]). An ideal I of A,h is said to be a kernel ideal if there is a congruence Ψ on A such that(2)I=Ψ0=a∈A:a,0∈Ψ.Definition 4.
(see [29]). A Heyting algebra is an algebra K,∨,∧,⟶,0,1 of type 2,2,2,0,0 where K,∨,∧,0,1 is a bounded distributive lattice and ⟶ is a binary operation on K such that for every a,b∈K, c≤a⟶b⇔c∧a≤b.Lemma 1.
(see [6]). Let A,h∈Kn,0 and a,b∈A. If a∨=∨k=0n−1h2ka and a∧=∧k=0n−1h2k+1a, then(1)
a≤b⇒a∨≤b∨(2)
a≤b⇒a∧≤b∧(3)
ha∨=b∧(4)
ha∧=a∨(5)
a∨∨=a∧∧=a∨(6)
a∨∧=a∧∨=a∧(7)
a∨b∨=a∨∨b∨(8)
a∨b∧=a∧∧b∧(9)
a∧b∧∧=a∧∨b∨(10)
a∧b∧∨=a∨∧b∧Throughout this article,L is a none trivial complete lattice satisfying infinite meet distributive law: a∧∧K=∨a∧x:x∈, ∀a∈A, and K⊆L. An L-fuzzy subset λ of a nonempty K is a mapping from K into L.In this work, for simplicity, we sayL-subsets instead of L-fuzzy subsets and write ζ∈LK to say that ζ is an L-subset of K.The union and intersection of any classλii∈Δ of L-subsets of K, respectively, represented by ∪i∈Δλi and ∩i∈Δλi, are defined as follows:(3)∪i∈Δλia=∨i∈Δλiaand∩i∈Δλia=∧i∈Δλia,for all a∈A, respectively.Definition 5.
(see [9]). For every λ and σ in LK, define a binary relation“⊆” on LK by(4)λ⊆σ⇔λa≤σa,∀a∈K.It can be easily proved that″⊆″ is a partial ordering on the set LK of L-subsets of K and the poset LK,⊆ forms a complete lattice in which for any λii∈Δ⊆LK,(5)∨i∈Δλi=∪i∈Δλiand∧i∈Δλi=∩i∈Δλi.The partial ordering″⊆” is called the point wise ordering.Forλ∈LK and α∈L, the set(6)λα=x∈K:λa≥α,is called the α-level subset of λ and for each a∈K, we have(7)λa=∨α∈L:a∈λα.For anyα∈L, we write α¯ to denote the constant L-subset of K which maps every element of K onto α.Definition 6.
(see [11]). Suppose h is a function from T into R, and suppose λ is an L-subset of T and σ is an L-subset of R. Then, the image of λ under h, hλ, is an L-subset of R defined as for each b∈R,(8)hλb=supλa:a∈h−1b,ifh−1b≠∅,0,otherwise.
The preimage ofσ under h, h−1σ, is an L-subset of T and h−1σa=σha for each a∈T.Definition 7.
(see [22]). An L-fuzzy subset ξ of a lattice K is called an L-fuzzy ideals of K if ξ0=1,ξx∨b≥ξx∧ξb, and ξx∧b≥ξx∨ξb.
AnL-fuzzy subset ξ of a lattice K is called an L-fuzzy filters of K, if ξ1=1,ξx∧b≥ξx∨ξb, and ξx∨b≥ξx∧ξb, for each x,y∈K.
It was also proved in [22] that an L-subset ξ of a lattice K with 0 is called an L-ideal of K if ξ0=1 and ξx∨y=ξx∧ξy for each x,y∈K. Dually, an L-subset ξ of a lattice K with 1 is called an L-filter of K if ξ1=1 and ξx∧y=ξx∧ξy, for each x,y∈K.AnL-ideal (respectively, L-filter) λ of K is called proper if it is not a constant map 1¯.Definition 8.
(see [28]). An L-subset λ of K is said to be an L-down set (respectively, L-up set) if a≤b, then λa≥λb (respectively, λa≤λb) for every a,b∈K.Lemma 2.
(see [28]). Let λ be an L-subset of K. Then, the L-subset λ↓ of K defined by(9)λ↓a=∨λb:a≤bfor everya∈K,is the least L-down set containing λ.Dually, we have the next result.Lemma 3.
(see [28]). Let λ be an L-subset of A. Then, the L-fuzzy subset λ↑ of A defined by(10)λ↑a=∨λb:b≤afor eacha∈A,is the least L-up set including λ.In what follows, for an Ockham algebraA, we shall denote by ℒℐKA the set of all kernel L-ideals of A and by ℒℐA the lattice of L-ideals of A in which the lattice operations ∧ and ∨ are given by(11)λ∧σ=λ∩σ;λ∨σa=∨μx∧σy:a=x∨y,for alla∈A.By anL-binary relation on a nonempty set K, we mean an L-subset of K×K. For an L-binary relation Ψ on K and each α∈L, the set(12)Ψα=a,b∈K×K:Ψa,b≥α,is called the α−level binary relation of Ψ on K.Definition 9.
(see [30]). An L-relation Ψ on a nonempty set K is said to be(1)
Reflexive ifΨa,a=1, for all a∈k(2)
Symmetric ifΨa,b=Ψb,a, for each a,b∈X(3)
Transitive if for eacha,b∈K: Ψa,b≥Ψa,c∧Ψc,b for all c∈K
AnL-equivalence relation on K is a reflexive, symmetric, and transitive L-relation on K.Definition 10.
(see [28]). Ψ is an L-equivalence relation on A,h and is called an L-congruence relation on A,h if it compatible with ∧,∨ and a unary operation h.For anya∈L and Ψ is L-congruence relation, L-subset Ψt of A,h is defined as follows:(13)Ψta=Ψt,a,for eacha∈A.We callΨt an L-congruence class of Ψ determined by t, and in particular, Ψ0 is called the kernel of Ψ and Ψ1 is called the cokernel of Ψ. One can easily observe that the kernel Ψ0 of Ψ is an L-ideal of A and the cokernel Ψ1 of Ψ is an L-filter of A.PutA/Ψ=Ψa:a∈A and ∧,∨ are binary operations and h is a unary operation on Α/Ψ expressed as follows:(14)Ψa∧Ψb=Ψa∧b,Ψa∨Ψb=Ψa∨bandhΨa=Ψha.After routine work, it can be proved thatA/Ψ,∧,∨,h,Ψ0,Ψ1 is an Ockham algebra and it is said to be the quotient Ockham algebra of A modulo Ψ.Definition 11.
(see [28]). An L-ideal λ of A,h is called a kernel L-ideal if λ=Ψ0 for some L-congruence Ψ of A.Lemma 4.
(see [28]). An L-ideal λ of A,h is a kernel L-ideal if and only if it holds the following conditions:(1)
h2λ⊆λ(2)
hλ↑s∧λa∧s≤λa, for each a,s∈ALemma 5.
(see [28]). The intersection of a class of kernel L-ideals of A,h is a kernel L-ideal.


## 3. KernelL-Ideals in a Subclass of Ockham Algebra

In the present topic, we present the structure of the set of kernelsL-ideals in a subclass Kn,0 of the class O of Ockham algebra.Lemma 6.
LetA,h∈Kn,0. Then, any kernel L-ideal of A is determined by an L-filter of A.Proof.
Suppose thatλ be a kernel L-ideal of A. This implies that there exists an L-congruence Ψ on A such that λ=kerΨ. Put γ=cokerΨ=Ψ1, which is an L-filter of A.
Consider anL-subset of A defined as follows:(15)γ0a=∨γx:a≤∨k=0n−1h2k+1x,x∈A,for alla∈A.
To determineλ by the L-filter γ of A, we want to show that λ=γ0. Now, for any a∈A,(16)λa=kerΨa=Ψa,0≤Ψha,1=γha.
Sincea≤∨k=0n−1h2ka=∨k=0n−1h2k+2a=∨k=0n−1h2k+1ha.
Now, we get that(17)λa=γha≤∨γha:a≤∨k=0n−1h2k+1ha=γ0a.
Therefore,λ⊆γ0.(18)Also for everya∈A,γ0a=∨γx:a≤∨k=0n−1h2k+1x,x∈A=∨Ψa,1:a≤∨k=0n−1h2k+1x,x∈A≤∨Ψ∨k=0n−1h2k+1x,0:x≤∨k=0n−1h2k+1x,x∈A≤Ψa∧∨k=0n−1h2k+1x,a∧0=Ψa,0=λa.
That is,γ0⊆λ and hence γ0=λ. Thus, the result holds.Next, we see an equivalent characterization of anL-fuzzy ideal of a Kn,0-algebra to get a kernel L-ideal.Lemma 7.
LetA,h∈Kn,0. Then, an L-ideal λ of A,h is a kernel L-ideal if and only if the following conditions are satisfied:(1)
λh2a≥λa, for each a∈A(2)
λx∧λa∧∧k=0n−1h2k+1x≤λa, for each a,x∈AProof.
Assume thatλ is a kernel L-ideal of A. Let a∈A. Then, by Lemma 4 (1), we have λa≤h2λh2a≤λh2a. Hence, (1) holds.
For eachx,a∈A,(19)λx∧λa∧∧k=0n−1h2k+1x≤λ∨k=0n−1h2kx∧λa∧∧k=0n−1h2k+1x≤hλ∧k=0n−1h2k+1x∧λa∧∧k=0n−1h2k+1x≤hλ∧k=0n−1h2k+1x∧λa∧∧k=0n−1h2k+1x≤λabyLemma2.152.
Thus, (2) is proved. Conversely, suppose that the given conditions hold. Let a∈A,(20)h2λa=∨λx:a=h2x≤∨λh2x:a=h2x=λa.
Hence,h2λ⊆λ. Again for every a,b∈A,(21)hλ↑b∧λa∧b=∨hλx:x≤b∧λa∧b=∨∨λy:hy=x:x≤b∧λa∧b≤∨λy:hy≤b∧λa∧b=∨λy∧λa∧b:hy≤b≤∨λy∧λa∧hy:hy≤b≤∨λy∧λa∧∧k=0n−1h2k+1y:hy≤b≤λaby2.
Then, it follows from Lemma4 that λ is a kernel L-ideal.In the following result, we characterized the least kernelL-ideal of A,h.Theorem 1.
LetA,h∈Kn,0 and λ be an L-ideal of A. Let λ° be an L-subset of A defined by(22)λ°a=∨λx:a∧x∧≤x∨∧x∧,foralla∈A,where x∨ and x∧ are as stated in Lemma 1. Then, λ° is the least kernel L-ideal of A containing λ.Proof.
First we prove thatλ° is an L-ideal of A.(23)λ°0=∨λx:0∧x∧≤x∨∧x∧=∨λx:0≤x∨∧x∧≥λ0=1.as0≤0∨∧0∧.
Hence,λ°0=1.
Now, for eacha,b∈A,(24)λ°a∧λ°b=∨λx:a∧x∧≤x∨∧x∧∧∨λy:b∧y∧≤y∨∧y∧=∨λx∧λy:a∧x∧≤x∨∧x∧,b∧y∧≤y∨∧y∧≤∨λx∨y:a∨b∧x∨y∧≤x∨y∨∧x∨y∧=λ°a∨b.
This implies thatλ°a∧λ°b≤λ°a∨b for each a,b∈A.
On the other side,(25)λ°a∨b=∨λx:a∨b∧x∧≤x∨∧x∧≤∨λx:a∧x∧≤x∨∧x∧=λ°a.
With similarly approach, we can prove thatλ°a∨b≤λ°b and so λ°a∨b≤λ°a∧λ°b. Therefore, λ°a∨b=λ°a∧λ°b and thus λ° is an L-ideal of A.
Next, we prove thatλ° is a kernel L-fuzzy ideal of A.(26)λ°a=∨λx:a∧x∧≤x∨∧x∧≤∨λh2x:h2a∧x∧≤h2x∨∧x∧=∨λh2x:h2a∧h2x∧≤h2x∨∧h2x∧∨λh2x:h2a∧x∧≤x∨∧x∧=λ°h2a.
Thus,λ°a≤λ°h2a, for each a∈A, and the property (1) of Lemma 7 holds.
To see the property (2) of Lemma 7 holds, let a,s∈A. Then,(27)λ°s∧λ°a∧∧k=0n−1h2k+1t=λ°s∧λ°a∧s∧=∨λx:s∧x∧≤x∨∧x∧∧∨λy:a∧s∧∧y∧≤y∨∧y∧=∨λx∧λy:s∧x∧≤x∨∧x∧,a∧s∧∧y∧≤y∨∧y∧≤∨λx∨y:a∧x∨y∧≤x∨y∧∧x∨y∨≤λ°y.
Hence, by Lemma7, λ° is a kernel L-ideal of A. Suppose λ is a kernel ideal of A such that λ⊆γ.
This implies there exists anL-congruence Ψ on A such that γ=kerΨ. Let x∈A. As a∧a∧≤a∨∧a∧, we clearly get(28)λ°a=∨λx:a∧x∧≤x∨∧x∧,≥λa.
And henceλ⊆λ°. Again for each a∈A,(29)λ°a=∨λx:a∧x∧≤x∨∧x∧≤∨γx:a∧x∧≤x∨∧x∧=∨Θx,0:a∧x∧≤x∨∧x∧≤∨Ψh2k+1x,1:a∧x∧≤x∨∧x∧≤∨Ψ∧k=0n−1h2k+1x,1:a∧x∧≤x∨∧x∧=∨Ψx∧,1:a∧x∧≤x∨∧x∧≤Ψa∧,1.
Similarly, we can show thatλ°a≤∨Θx∨,0:a∧x∧≤x∨∧x∧.
This implies that(30)λ°a≤∨Ψx∧,1:a∧x∧≤x∨∧x∧∧∨Ψx∨,0:a∧x∧≤x∨∧x∧=∨Ψx∧,1∧Ψx∨,0:a∧x∧≤x∨∧x∧=∨Ψa∧x∧,a∧Ψx∨∧x∧,0:a∧x∧≤x∨∧x∧≤Ψx∨∧x∧,a∧Ψx∨∧x∧,0≤Ψa,0=γa.
And soλ°⊆γ. Therefore, from these observations, λ° is the least kernel L-ideal of A including λ.The following corollaries follows Theorem1.Corollary 1.
SupposeA,h∈Kn,0 and λ is an L-ideal of A. Then, λ is a kernel L-ideal if and only if λ=λ°.Corollary 2.
IfA,h∈Kn,0, then the following properties hold, for all λ,γ∈ℒℐA:(1)
λ⊆γ⇒λ°⊆γ°(2)
λ⊆λ°=λ°°(3)
λ°∨γ°⊆λ∨γ°(4)
λ∩γ°⊆λ°∩γ°Given aKn,0- algebra A and L-ideals λ, γ of A, we shall define(31)λ°⊓γ°=λ∩γ°andλ°⊔γ°=λ∨γ°.Suppose thatλ and γ kernel L-ideals of A, then λ⊓γ=λ∩γ. Indeed, since λ and γ are kernel L-ideals of A, we have λ=λ° and γ=ν°, and λ∩γ is also a kernel L-ideal. Thus,(32)λ⊓γ=λ°⊓γ°=λ∩γ°=λ∩γ.Then, we have the following.Theorem 2.
LetA,h∈Kn,0 and ℒℐkA denote the set of all kernel L-ideals. Then, ℒℐkA,⊓,⊔ is a complete bounded distributive lattice.Proof.
Suppose thatλ,γ∈ℒℐkL. Then, λ⊓γ=λ∩γ is a kernel L-ideal and the infimum of λ and γ. It follows by Theorem 1 that λ⊔γ=λ°⊔γ°=λ∨γ° is a kernel L-ideal and λ,γ⊆λ∨γ⊆λ∨γ°. Suppose that ξ is a kernel L-ideal of A such that ξ is an upper bounded of λ and γ, then λ∨γ⊆ξ and there follows λ∨γ°⊆ξ°=ξ. Hence, λ∨γ° is the supremum of both λ and γ in ℒℐKA. Thus, ℱℐKA,⊓,∩ is a lattice. Obviously, χ0 and χL are the least and biggest kernel L-ideals in ℒℐkA, respectively. This implies that ℒℐkA is bounded. The completeness is clear since the intersection of a family of kernel L-ideals is also a kernel L-ideal of A (Lemma 4). As far as the distributivity, let λ,γ,ξ∈ℒℐkA. Since(33)λ⊓γ⊔ξ=λ°⊓γ°⊔ξ°=λ°⊓γ∨ξ°=λ∩γ∨ξ°=λ∩ξ∨γ∨λ°=λ∩ξ°∨γ∧ξ°=λ°⊓ξ°∨γ∧ξ°=ξ∩ξ°⊔γ°⊓ξ°=λ⊓ξ⊔γ⊓ξ.
It follows thatℒℐkA is a distributive lattice.In order to further characterize the structure of the lattice of kernelL-ideals of a Kn,0-algebra, we require the following:Theorem 3.
LetA,h∈Kn,0. Then, ℒℐkA,⊓,⊔,∗ is a Heyting algebra in which for λ,γ∈ℒℐkA, the relative pseudocomplement of λ and γ is defined as for every a∈A,(34)λ∗γa=∨ξa∨:ξ∈ℒℐA,ξ°⊓λ⊆γ.Proof.
Sinceγ°⊓λ⊆γ and(35)γ0∨=γ∨k=0n−1h2k0=∧k=0n−1γh2k0≥γ0=1,we have λ∗γ0≥γ0∨=1.(36)λ∗γa=∨ξa∨:ξ∈ℱℐA,ξ°⊓λ⊆γ≤∨ξa∨b∨:ξ∈ℱℐAξ°⊓λ⊆γ=λ∗γa∨b.
Similarly, we can prove thatλ∗γb≤λ∗γa∨b and hence(37)λ∗γa∨λ∗γb≤λ∗γa∨b.
On the other side, for eacha,b∈A,(38)λ∗γa∧λ∗γb=∨ξa∨:ξ∈ℱℐA,ξ°⊓λ⊆γ∧∨σy∨:σ∈ℱℐA,σ°⊓λ⊆γ=∨ξa∨∧σb∨:ξ°⊓λ⊆γ,σ°⊓λ⊆γ≤∨ξ⊔σa∨∧ξ⊔σb∨:ξ⊔σ°⊓λ⊆γ≤∨ξ⊔σa∨b∨:ξ⊔σ°⊓λ⊆γ=λ∗γa∨b.
Thus,λ∗γa∨b=λ∗γa∧λ∗γb, for each a,b∈A, and hence λ∗γ is an L-ideal of A.
Next, we show thatλ∗γ is a kernel L-ideal of A.(39)λ∗γa=∨ξa∨:ξ∈ℱℐA,ξ°⊓λ⊆γ=∨γf2a∨:γ∈ℱℐA,γ°⊓λ⊆γ=λ∗γf2a∨.
Also, for anya,s∈A,(40)λ∗γs∧λ∗γa∧t∧=∨ξs∨:ξ∈ℱℐA,ξ°⊓λ⊆γ∧∨δa∧s∧∨:δ∈ℱℐA,δ°⊓λ⊆γ=∨ξs∨∧δa∨∧s∧:ξ°⊓λ⊆γ,δ°⊓λ⊆γ≤∨ξ∨δs∨∧ξ∨δa∨∧s∧:ξ°⊓λ⊆γ,δ°⊓λ⊆γ≤∨ξ∨δs∨∨a∨∧s∧:ξ∨δ°⊓λ⊆γ≤∨ξ∨δa∨:ξ∨δ°⊓λ⊆γ≤λ∗γa.
Hence, by Lemma4, λ∗γ is kernel L-ideals of A.
Next, we prove thatμ∗λ is the biggest kernel L-ideals of A such that λ⊓λ∗γ⊆ξ. Suppose that σ∈ℒℐKA such that λ⊓σ⊆ξ. Now, since for each a∈A,(41)σa≤σa∨andσ°⊓λ=σ°⊓λ°=σ∩λ°=σ⊓λ⊆γ.
We have clearly(42)λ∗γa=∨ξa∨:ξ∈ℒℐA,ξ°⊓λ⊆γ≥σa∨≥σa.
And henceσ⊆λ∗γ. Therefore, λ∗γ is the pseudocomplement of λ relative to γ.Theorem 4.
LetA,h∈Kn,0 and λ be a proper kernel L-ideal of A and L=0,1. Then, λ is the intersection of a family of prime kernel L-ideals including it.Proof.
Assume thatλ is a proper L-ideal of A. Then, there exists x∈A such that λx≠1. Put λx=α and consider the set(43)Δ=σ∈ℒℐKA:σ⊆λandσx≤α.
Clearly,λ∈Δ and so Δ is nonempty and hence it forms a poset under the point wise ordering “⊆”. By using Zorn’s lemma, we can choose a maximal element, say ξ in Δ; we prove that ξ is a prime kernel L-ideal. Let γ and σ be L-ideals of A such that γ∩σ⊆ξ. Suppose on contrary that γ⊈ξ and σ⊈ξ. Then,(44)ξ⊂ξ∨γ⊆ξ∨γ°andξ⊂ξ∨σ⊆σ∨ξ°.
By the maximality ofξ in Δ, it follows that γ∨ξ°∉Δ and σ∨ξ°∉Δ. So that ν∨ξ°a≰α and σ∨ξ°a≰α. Now,(45)ξ∨γ°⊓ξ∨σ°=ξ∨γ∩ξ∨σ°=ξ∨γ∩σ°⊆ξ°=ξ,which is a contradiction.
Letξii∈Δ be the set of prime kernel L-ideals that contain λ. Then, λ⊆∩i∈Δξi. Suppose that ∩i∈Δξi⊈λ. Then, there is x∈A such that ∩i∈Δξix≰λx. This implies that λx≠1 and hence λ is proper. Thus, from the above, there exists a prime kernel L-ideal ξ which contains λ and ξ≤α=λx and hence ∩i∈Δξix≤ξx≤α=λx, which is a contradictions. Hence, ∩i∈Δξi⊆λ and so λ=∩i∈Δξi.


## 4. L-Congruences of aKn,0-Algebra

In this topic, we characterize an interesting property of the leastL-congruence on the Kn,0-algebra A such that the given kernel L-ideal ξ of A as its L-congruence class.Letξ be a kernel L-ideal of A,h. As shown in [28], the smallest L-congruence ΩLξ on A with kernel ξ which is given by: for x,y∈A,(46)ΩLξx,y=∨ξa∧hξb:x∨a∧b=y∨a∧b.Theorem 5.
IfL,h∈Kn,0 and ξ,λ∈ℒℐKA, then(47)ΩLξ⊔λ=ΩLξ∨ΩLλ.Proof.
Letξ,λ∈ℒℐkA. It is easily proved that(48)ξ⊆λ⇔ΩLξ⊆ΩLλ.
Then, we haveΩLξ∨ΩLλ≤ΩLξ∨λ0=ΩLξ⊔λ. On the other side, let x,y∈A. Then, by (46), we have(49)ΩLξ⊔λx,y=∨μ⊔λr∧hξ⊔λt:x∨r∧t=y∨r∧t.
Consider the following equation:(50)ξ⊔λr=ξ∨λ0r=∨ξ∨λa:r∧a∧≤a∧∧a∨=∨ξ∨λa:r∨a∨∧a∧=a∧∧a∨=∨∨ξr1∧λt1:a=r1∨t1:r∨a∨∧a∧=a∧∧a∨.(51)hξ⊔λt=∨ξ∨λ0b:hb≤t=∨∨ξ∨λc:b∧c∧≤c∧∧c∨:hb≤t=∨∨ξ∨λc:b∨c∨∧c∧=c∧∧c∨:hb≤t=∨∨∨ξr2∧λt2:c=r2∨t2:b∨c∨∧c∧=c∧∧c∨:hb≤t=∨ξr2∧λt2:c=r2∨t2,b∨c∨∧c∧=c∧∧c∨,hb≤t=∨ξr2∧λt2:b∨r2∨∨t2∨∧r2∧∧t2∧=r2∨∨t2∨∧r2∧∧t2∧,hb≤t≤∨ξr2∧λt2:hb∧r2∧∧t2∧∨r2∨∨t2∨=r2∧∧t2∧∨r2∨∨t2∨,hb≤t.
By (49) and (50), we have(52)ΩLξ⊔λx,y≤∨ξ⊔λr:x∨r∧t=y∨r∧t=∨ξr1∧λt1:r∨r1∨∨t1∨∧r1∧∧t1∨=r1∨∨t1∨∧r1∧∧t1∨x∨r∧t=y∨r∧t,.
From (51), we have(53)ξr1≤ΩLξ0,r1≤ΩLξ0,r1∨∧ΩLξ1,r1∧≤ΩLξr,r∨r1∨∧ΩLξr∨r1∨∨t1∨,r∨r1∨∨t1∨∧r1∧,(54)ΩLξ1,r1∧=ΩLξ1,r1∧∧ΩLξr1∨∨t1∨,r1∨∨t1∨≤ΩLξr1∨∨t1∨,r1∨∨t1∨∧r1∨.
Also, from (51), we have(55)λt1≤ΩLλ0,t1≤ΩLλ0,t1∨∧ΩLλ1,t1∧=ΩLλ0,t1∨∧ΩLλr∨r1∨,r∨r1∨∧ΩLλ1,t1∧∧ΩLλr∨r1∨∨t1∨∧r1∧,r∨r1∨∨t1∨∧r1∧≤ΩLλr∨r1∨,r∨r1∨∨t1∨∧ΩLλ1,t1∧∧ΩLλr∨r1∨∨t1∧∧r1∧,r∨r1∨∨t1∨∧r1∧∧t1∧,(56)ΩLλt1∧,1=ΩLλt1∧,1∧ΩLλr1∨∨t1∨,r1∨∨t1∨≤ΩLλr1∨∨t1∨∧r1∧,r1∨∨t1∨∧r1∧≤ΩLλr1∨∨t1∨∧r1∧∧t1∧,r1∨∨t1∨∧r1∧.
From (52), (53), (54), (55), and (56), we have(57)ΩLξ⊔λx,y≤∨ΩLξ∨ΩLλr,0.
Similarly,(58)ΩLξ⊔λx,y≤∨ΩLξ∨ΩLλt,1.
From (57) and (58), we get(59)ΩLξ⊔λx,y≤ΩLξ∨ΩLλx,y.
Hence,ΩLξ⊔λ=ΩLξ∨ΩLλ.We now give a description on the biggestL-congruence on a Kn,0-algebra A such that the given kernel L-ideal of A as its L-congruence class.Theorem 6.
LetA,h∈Kn,0 and ξ be a kernel L-ideal of A. A binary relation δξ on A is defined as follows: δξa,b=∨ξhKa∧x∧ξhKb∧x for all x∈A and k=0,1,2,2n−1. Then, δξ is the biggest congruence on A with δξ=ξ.


## 5. Conclusion

In this work, we studied KernelL-ideals and L-congruence on a subclass of Ockham algebras and investigate their properties. We proved that the set of kernel L-ideal ℱℐkA of Kn,0-algebra A forms Heyting algebras. Also, we obtain the least, respectively, the biggest L-congruences on Kn,0-algebra A having a given L-ideal as a kernel and describe it using algebraic operations in an L setting.



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*Source: 7668044-2022-05-29.xml*