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

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra (A, 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.


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][4][5][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. e 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][9][10], some generalizations of fuzzy ideals in distributive lattices [11][12][13], fuzzy ideals and fuzzy filters of partially ordered sets [14,15], and fuzzy ideals of universal algebras [16][17][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. e notions of fuzzy congruence relations were studied in various algebraic structures: in semigroups (see [23,24]), in groups, rings, and semirings (see [25][26][27][28][29][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][38][39]). e 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. e 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: 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.

Preliminaries
is section contains some basic definitions and results which will be used the sequel.
Definition 2 (see [42]). A congruence relation on an Ock- Definition 3 (see [41]). By an ideal of an Ockham algebra (A, 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 By an L-fuzzy subset μ of a nonempty X, we mean a mapping from X into L. e set of all L-fuzzy subsets of X is denoted by L X .
Definition 4 (see [43]). Let μ, σ in L X . en, the Cartesian product of μ and σ, denoted by μ × σ, is defined by, for all x, y ∈ X, e union and intersection of any family μ i i∈Δ of L-fuzzy subsets of X, respectively, denoted by ∪ i∈Δ μ i and ∩ i∈Δ μ i , are defined by for all x ∈ X, respectively.
Definition 5 (see [44]). For any μ and σ in L X , define a binary relation " ⊆ " on L X by μ ⊆ σ if and only if μ(x) ≤ σ(x), for all x ∈ X. (6) It can be easily verified that ⊆ is a partial order on the set L X of L-fuzzy subsets of X and the poset (L X , ⊆ ) forms a complete lattice, in which, for any μ i i∈Δ ⊆ L X , e partial ordering " ⊆ " is called the pointwise ordering.
For μ ∈ L X and α ∈ L, the set, is called the α-level subset of μ, and for each x ∈ X, we have For any α ∈ 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]). Let f be a function from X into Y, and let μ be an L-fuzzy subset of X. en, the image of μ under f, denoted by f(μ), is an L-fuzzy subset of Y given by, for all y ∈ Y, e preimage of σ under f, symbolized by f − 1 (σ), is an L-fuzzy subset of X and Definition 7 (see [33]). An L-fuzzy subset μ of a lattice X with 0 is said to be an L-fuzzy ideal of X if μ(0) � 1 and Dually, an L-fuzzy subset μ of a lattice X with 1 is said to be an L-fuzzy filter of X if μ(1) � 1 and An L-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, is called the α-level binary relation of Θ on X.
Definition 8 (see [45]). An L-fuzzy relation Θ on a nonempty set X is said to be A reflexive, symmetric, and transitive L-fuzzy relation on X is called an L-fuzzy equivalence relation on X.

L-Fuzzy Congruences of Ockham Algebras
In this section, we give various characterizations of an L-fuzzy congruence relation of an Ockham algebra. roughout this section and the rest, A stands for an Ockham algebra (A, f).

Definition 9.
An L-fuzzy equivalence relation Θ on A is called an L-fuzzy congruence relation on A if it satisfies the following conditions: An L-fuzzy equivalence relation on A that satisfies condition (1) is called a lattice L-fuzzy congruence on A. e following two lemmas give important characterization for L-fuzzy congruence relations in Ockham algebras.

Lemma 2. An L-fuzzy equivalence relation Θ on A is an L-fuzzy congruence relation on A if and only if, for any
For any a ∈ L and L-fuzzy congruence relation Θ, define an L-fuzzy subset Θ a of A by We call Θ a an L-fuzzy congruence class of Θ determined by a, and in particular, Θ 0 is called the kernel of Θ. One can easily observe that kernel Θ 0 of Θ is an L-fuzzy ideal of A.

Lemma 3. Let Θ be a fuzzy congruence on A.
For any x, y ∈ A, the following holds } and define binary operations ∧ , ∨ and a unary operation f on A/Θ by It is routine to verify that ((A/Θ), ∧ , ∨ , f, Θ 0 , Θ 1 ) 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 Θ L (μ) (respectively, (Θ lat ) L (μ)) 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, whenever μ is an L-fuzzy ideal of A and whenever μ is an L-fuzzy filter of A.
For a given L-fuzzy ideal μ of A, we shall now investigate the smallest L-fuzzy congruence Θ L (μ) of A containing μ × μ.
is the smallest L-fuzzy down set containing μ.
erefore, 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 □ Dually, we have the following lemma.
for all x, y ∈ A is an L-fuzzy congruence relation on A.
Next, we show that Ω L (μ) satisfies the substitution properties. If , Similarly, we can show that Now, us, Ω L (μ) is an L-fuzzy congruence on A.
□ Theorem 1. For any L-fuzzy ideal μ of A, where the join is taken in the lattice of L-fuzzy congruences on A.

Journal of Mathematics
, is implies that

L-Fuzzy Kernel Ideals
In this section, we study L-fuzzy kernel ideals in Ockham algebras and give several characterizations for them.
e following observation is immediate.

Lemma 9.
If μ is an L-fuzzy ideal of A such that f 2 (μ) ⊆ μ, then, for every n, from which it follows that e following lemma gives an internal characterization for L-fuzzy kernel ideals in Ockaham algebras.

Corollary 2. If μ is a proper kernel L-fuzzy ideal of
Proof. Suppose that μ is a proper kernel L-fuzzy ideal of A. en, μ � Θ 0 , for some L-fuzzy congruence Θ of A and μ(a) ≠ 1, for some a ∈ A. Put α � μ(a). Now, we claim that en, the claim clearly holds if Hence, the claim is true.

Lemma 11. Arbitrary intersection of a family of kernel L-fuzzy ideals of A is a kernel L-fuzzy ideal.
We denote the set of all kernel L-fuzzy ideals of A by FI K (A).

Theorem 2.
e set FI K (A) of kernels L-fuzzy ideal of A ordered by set inclusion forms a complete lattice.

Theorem 3.
(1) FI 2 (A) is a complete sublattice of FI(A) (2) If μ is a kernel L-fuzzy ideal of A, then μ ∈ FI 2 (A) Proof. Let μ i : i ∈ Δ be a nonempty subset of FI 2 (A).
Now, suppose that (f 2 ) −1 (x) ≠ ∅. en, Hence, the claim is true. Since χ A is greatest element of FI 2 (A), we have FI 2 (A) is a complete sublattice of FI(A).
It follows from Lemma 10.
is an L-fuzzy ideal of A containing μ.
Proof. Suppose that μ be an L-fuzzy ideal of A. Now, since We have μ(0) � 1. Let x, y ∈ A. en,

□
In the following theorem, we give a necessary and sufficient condition for an L-fuzzy ideal of an Ockham algebra to be a kernel L-fuzzy ideal. Proof. Suppose that μ is a kernel L-fuzzy ideal of A. en, by Lemmas 10 and 12, we have f 2 (μ) ⊆ μ and μ ⊆ μ. Now, since, for any x ∈ A, we have μ ⊆ μ. Hence, f 2 (μ) ⊆ μ and μ � μ. Conversely, suppose that μ is an L-fuzzy ideal of A satisfying the given condition. en, by Lemma 11, it is enough to show that Let x, t ∈ A if ((f 2 ) −1 (t)) � ∅; clearly, (57) holds. Suppose that ((f 2 ) −1 (t)) ≠ ∅. en, Hence, μ is a kernel L-fuzzy ideal of A.

Congruences with a Given L-Fuzzy Kernel Ideal
In this section, we describe the smallest and the largest L-fuzzy congruences on an Ockham algebra having a given L-fuzzy kernel ideal. From Lemmas 8 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 Ω L (μ) which is given by We can in fact establish the following simpler version. Theorem 6. If μ ∈ FI K (A), then the smallest L-fuzzy congruence on A with kernel μ is the L-fuzzy relation Φ L (μ) is given by Proof. To prove it enough to show that Φ L (μ) � Ω L (μ), since μ(i) ≤ f −1 (f(μ))(i) � f(μ)(f(i)), we have ≤ Ω L (μ)(x, y).
Since μ is kernel L-fuzzy ideal,
Having described the smallest L-fuzzy congruence on A with a given kernel L-fuzzy ideal, we now proceed to determine the largest such L-fuzzy congruence.
For a given L-fuzzy ideal μ of A, each a ∈ A, and for each n ∈ N, consider the L-fuzzy subset of A defined by and an L-fuzzy binary relation Θ μ n on A defined by, for all a, b ∈ AΘ μ n (a, b) � sup W μ a,n (x) ∧ W μ b,n (x): x ∈ A .
(67) Lemma 13. Θ μ n is a lattice L-fuzzy congruence of A and ∩ n∈N Θ μ n is an L-fuzzy congruence of A.