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

In this paper, we study L -congruences and their kernel in a subclass K n, 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.


Introduction
e concept of an Ockham algebra was rst 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 K n,0 of the variety of Ockham Algebras (in which h 2n id A ). e 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 rst time, the concept of fuzzy sets was presented by Zadeh as an extension of the classical notion of set theory [8]. He de ned 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 de nition of fuzzy subsets. Swamy and Swamy [10] studied that complete lattices that ful ll the in nite meet distributive law are the most appropriate candidates to have the truth values of general fuzzy statements. e study of fuzzy subalgebras of di erent algebraic structures has been begun after Rosenfeld presented his paper [11] on fuzzy subgroups. is paper has provided su cient motivation to researchers to study the fuzzy subalgebras of di erent 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. e concept of fuzzy congruence relations was presented in di erent algebraic structure: in semigroups (see [12,13]), in groups, semirings, and rings (see [14][15][16][17][18][19]), in modules and vector spaces (see [20,21]), in lattices (see [22,23]), in universal algebras (see [24,25]), and more recently in MSalgebras and Ockham Algebras (see [26][27][28]). Initiated by the above results, we present Kernel L-ideals and L-Congruence on a subclass FI k (A) of K n,0 of the variety of Ockham algebras and study their characteristics. We prove that the class of kernel L-ideal LI k (A) of K n,0 -algebra A forms Heyting algebras. Also, we get the least and the biggest L-congruences, respectively, on K n,0 -algebra A having a given L-ideal as an L-kernel.

Preliminaries
is section contains basic de nitions and important results which will be used in the sequel.
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) 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 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.
roughout 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 say L-subsets instead of L-fuzzy subsets and write ζ ∈ L K to say that ζ is an L-subset of K.
e union and intersection of any class λ i i∈Δ of L-subsets of K, respectively, represented by ∪ i∈Δ λ i and ∩ i∈Δ λ i , are defined as follows: for all a ∈ A, respectively.
Definition 5 (see [9]). For every λ and σ in L K , define a binary relation"⊆" on L K by It can be easily proved that ″ ⊆ ″ is a partial ordering on the set L K of L-subsets of K and the poset (L K , ⊆) forms a complete lattice in which for any λ i i∈Δ ⊆L K , e partial ordering ″ ⊆" is called the point wise ordering.
For λ ∈ L K and α ∈ L, the set is called the α-level subset of λ and for each a ∈ K, we have For any α ∈ 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.
Definition 7 (see [22]). An L-fuzzy subset ξ of a lattice K is called an An L-fuzzy subset ξ of a lattice K is called an L-fuzzy It was also proved in [22] that an L-subset ξ of a lattice K with 0 is called an 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. en, the is the least L-up set including λ.
In what follows, for an Ockham algebra A, we shall denote by LI K (A) the set of all kernel L-ideals of A and by LI(A) the lattice of L-ideals of A in which the lattice operations ∧ and ∨ are given by By an L-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 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 An L-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 any a ∈ L and Ψ is L-congruence relation, L-subset Ψ t of (A, h) is defined as follows: We call Ψ t an L-congruence class of Ψ determined by t, and in particular, Ψ 0 is called the kernel of Ψ and Ψ 1 is called the cokernel of Ψ. One can easily observe that the kernel Ψ 0 of Ψ is an L-ideal of A and the cokernel Ψ 1 of Ψ is an L-filter of A.
{ } and ∧, ∨ are binary operations and h is a unary operation on A/Ψ expressed as follows: � Ψ(h(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: for each a, s ∈ A Lemma 5 (see [28]). e intersection of a class of kernel L-ideals of (A, h) is a kernel L-ideal.

Kernel L-Ideals in a Subclass of Ockham Algebra
In the present topic, we present the structure of the set of kernels L-ideals in a subclass K n,0 of the class O of Ockham algebra.

Lemma 6
Let (A, h) ∈ K n,0 . en, any kernel L-ideal of A is determined by an L-filter of A.
Proof. Suppose that λ be a kernel L-ideal of A. is implies that there exists an L-congruence Ψ on A such that λ � ker Ψ. Put c � coker Ψ � Ψ 1 , which is an L-filter of A.
Consider an L-subset of A defined as follows: To determine λ by the L-filter c of A, we want to show that λ � c 0 . Now, for any a ∈ A, at is, c 0 ⊆λ and hence c 0 � λ. us, the result holds. □ Next, we see an equivalent characterization of an L-fuzzy ideal of a K n,0 -algebra to get a kernel L-ideal.
en, it follows from Lemma 4 that λ is a kernel L-ideal.

□
In the following result, we characterized the least kernel L-ideal of (A, h).

Theorem 1. Let (A, h) ∈ K n,0 and λ be an L-ideal of A. Let λ°be an L-subset of
where x ∨ and x ∧ are as stated in Lemma 1. en, λ°is the least kernel L-ideal of A containing λ.
Proof. First we prove that λ°is an L-ideal of A.
To see the property (2) of Lemma 7 holds, let a, s ∈ A. en, Hence, by Lemma 7, λ°is a kernel L-ideal of A. Suppose λ is a kernel ideal of A such that λ⊆c.
is implies there exists an L-congruence Ψ on A such that c � ker Ψ. Let x ∈ A. As a∧a ∧ ≤ a ∨ ∧a ∧ , we clearly get And hence λ⊆λ°. Again for each a ∈ A, Similarly, we can show that λ°(a) ≤ ∨ Θ(x ∨ , 0): And so λ°⊆c. erefore, from these observations, λ°is the least kernel L-ideal of A including λ. □ e following corollaries follows eorem 1.   Given a K n,0 -algebra A and L-ideals λ, c of A, we shall define λ°⊓c°� (λ ∩ c)°and λ°⊔c°� (λ∨c)°. (31) Suppose that λ and c kernel L-ideals of A, then λ⊓c � λ ∩ c. Indeed, since λ and c are kernel L-ideals of A, we have λ � λ°and c � ]°, and λ ∩ c is also a kernel L-ideal. us, en, we have the following. is implies that LI k (A) is bounded. e 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, c, ξ ∈ LI k (A). Since It follows that LI k (A) is a distributive lattice.

□
In order to further characterize the structure of the lattice of kernel L-ideals of a K n,0 -algebra, we require the following: Theorem 3. Let (A, h) ∈ K n,0 . en, (LI k (A), ⊓, ⊔, * ) is a Heyting algebra in which for λ, c ∈ LI k (A), the relative pseudocomplement of λ and c is defined as for every a ∈ A, Proof. Since c°⊓λ⊆c and Similarly, we can prove that (λ * c)(b) ≤ (λ * c)(a∨b) and hence On the other side, for each a, b ∈ A, us, (λ * c)(a∨b) � (λ * c)(a)∧(λ * c)(b), for each a, b ∈ A, and hence λ * c is an L-ideal of A.
Next, we show that λ * c is a kernel L-ideal of A.
Also, for any a, s ∈ A, 6
erefore, λ * c is the pseudocomplement of λ relative to c. Proof. Assume that λ is a proper L-ideal of A. en, there exists x ∈ A such that λ(x) ≠ 1. Put λ(x) � α and consider the set 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 c and σ be L-ideals of A such that c ∩ σ⊆ξ. Suppose on contrary that c⊈ξ and σ⊈ξ. en, which is a contradiction.

L-Congruences of a K n,0 -Algebra
In this topic, we characterize an interesting property of the least L-congruence on the K n,0 -algebra A such that the given kernel L-ideal ξ of A as its L-congruence class.

□
We now give a description on the biggest L-congruence on a K n,0 -algebra A such that the given kernel L-ideal of A as its L-congruence class. for all x ∈ A and (k � 0, 1, 2, 2n − 1). en, δ(ξ) is the biggest congruence on A with δ(ξ) � ξ.

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

Data Availability
No data were used to support the results of this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.