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 , 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 to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on having a given fuzzy ideal as its kernel.
Zadeh  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  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  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 , Atanassov  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 , B-algebras , and BCK-algebras  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  in 1977. In simple terminology, an Ockham algebra is a bounded distributive lattice equipped with a dual endomorphism. Blyth and Silva  have studied and characterized kernel ideals in Ockham algebra. The purpose of this paper is to apply the theory of -fuzzy sets in the class of Ockham algebras, where is a complete lattice satisfying the infinite meet distributive law:for any and . To be specific, we study -fuzzy congruences and -fuzzy kernel ideals of Ockham algebras and investigate their properties. We also derive some equivalent conditions for every -fuzzy ideal of an Ockham algebra to become an -fuzzy kernel ideal. We give an internal characterization for the smallest and the largest -fuzzy congruences on having a given -fuzzy ideal as a kernel.
This section contains some basic definitions and results which will be used the sequel.
Definition 1 (see ). An Ockham algebra is an algebra of type in which is a bounded distributive lattice and is a unary operation on such that and the following de Morgan laws hold
For simplicity, we denote an Ockham algebra by a pair .
Definition 2 (see ). A congruence relation on an Ockham algebra is a lattice congruence on such that .
Definition 3 (see ). By an ideal of an Ockham algebra , we mean an ideal of as a distributive lattice. Moreover, an ideal of an Ockham algebra is called a kernel ideal if there exists a congruence on such that
By an -fuzzy subset of a nonempty , we mean a mapping from into . The set of all -fuzzy subsets of is denoted by .
Definition 4 (see ). Let in . Then, the Cartesian product of and , denoted by , is defined by, for all ,
The union and intersection of any family of -fuzzy subsets of , respectively, denoted by and , are defined byfor all , respectively.
Definition 5 (see ). For any and in , define a binary relation “” on by
It can be easily verified that is a partial order on the set of -fuzzy subsets of and the poset forms a complete lattice, in which, for any ,
The partial ordering “” is called the pointwise ordering.
For and , the set,is called the -level subset of , and for each , we have
For any , we write to denote the constant -fuzzy subset of which maps every element of onto .
Definition 6 (see ). Let be a function from into , and let be an -fuzzy subset of . Then, the image of under , denoted by , is an -fuzzy subset of given by, for all ,The preimage of under , symbolized by , is an -fuzzy subset of and
Definition 7 (see ). An -fuzzy subset of a lattice with is said to be an -fuzzy ideal of if and , for all .
Dually, an -fuzzy subset of a lattice with is said to be an -fuzzy filter of if and , for all .
An -fuzzy ideal (respectively, filter) of is said to be proper if it is not a constant map . By an -fuzzy binary relation on a nonempty set , we mean an -fuzzy subset of . For an -fuzzy binary relation on and each , the set,is called the -level binary relation of on .
Definition 8 (see ). An -fuzzy relation on a nonempty set is said to be(1)Reflexive if , for all (2)Symmetric if , for all (3)Transitive if, for each , , for all A reflexive, symmetric, and transitive -fuzzy relation on is called an -fuzzy equivalence relation on .
3. L-Fuzzy Congruences of Ockham Algebras
In this section, we give various characterizations of an -fuzzy congruence relation of an Ockham algebra. Throughout this section and the rest, stands for an Ockham algebra .
Definition 9. An -fuzzy equivalence relation on is called an -fuzzy congruence relation on if it satisfies the following conditions:(1)(2), for all
An -fuzzy equivalence relation on that satisfies condition (1) is called a lattice -fuzzy congruence on . The following two lemmas give important characterization for -fuzzy congruence relations in Ockham algebras.
Lemma 1. An -fuzzy relation on is an -fuzzy congruence relation on if and only if is a congruence relation on , for all .
Lemma 2. An -fuzzy equivalence relation on is an -fuzzy congruence relation on if and only if, for any ,
For any and -fuzzy congruence relation , define an -fuzzy subset of A by
We call an -fuzzy congruence class of determined by , and in particular, is called the kernel of . One can easily observe that kernel of is an -fuzzy ideal of .
Lemma 3. Let be a fuzzy congruence on . For any , the following holds(1) if and only if (2)Either or there exists such that
Let us put and define binary operations and a unary operation on by
It is routine to verify that is an Ockham algebra, and it is called the quotient Ockham algebra of modulo . For an -fuzzy subset of , we write (respectively, ) to denote the smallest -fuzzy congruence (respectively lattice -fuzzy congruence) on containing . It was proved in  that, for any ,whenever is an -fuzzy ideal of andwhenever is an -fuzzy filter of .
For a given -fuzzy ideal of , we shall now investigate the smallest -fuzzy congruence of containing .
Definition 10. An -fuzzy subset of is called an -fuzzy down set (respectively, -fuzzy up set) if, for any , (respectively, ) whenever .
Lemma 4. Let be an -fuzzy subset of . Then, the -fuzzy subset of defined byis the smallest -fuzzy down set containing .
Proof. Let . Since , we clearly haveThus, . Let such that . Now,Therefore, is an -fuzzy down set of . Let be any -fuzzy down set of such that . For any , we haveThis implies that . Therefore, is the smallest -fuzzy down set of containing .
Dually, we have the following lemma.
Lemma 5. Let be an -fuzzy subset of . Then, the -fuzzy subset of defined byis the smallest -fuzzy up set containing .
Lemma 6. Let be an -fuzzy ideal of . For each nonnegative integer , define . Then, is an -fuzzy ideal and is an -fuzzy filter of .
Proof. Now, sinceWe have .
Again, let . Then, if , or , , then we haveIf , and , , we haveAgain, as is an -fuzzy down set, we clearly have , and hence, . Therefore, is an -fuzzy ideal of . Analogously, we can prove that is an -fuzzy filter of .
Lemma 7. Let be any -fuzzy ideal of . Put and . Then,
Proof. Let . If , , we haveIf , , thenSimilarly, we can show that .
Lemma 8. Let be an -fuzzy ideal of . Then, the -fuzzy binary relation on defined byfor all is an -fuzzy congruence relation on .
Proof. Clearly, is reflexive and symmetric. We show that it is transitive. Let . If , then it can be easily verified that . Thus, we haveTherefore, is transitive, and hence, it is an - fuzzy equivalence relation on .
Next, we show that satisfies the substitution properties. If , then, after routine work, we obtainSimilarly, we can show thatFinally, if , then it can be easily verifiedNow,Thus, is an -fuzzy congruence on .
Theorem 1. For any -fuzzy ideal of ,where the join is taken in the lattice of -fuzzy congruences on .
Proof. We first show that and . Now, for any , we haveHence, . Similarly, we have . This implies thatTo establish the reverse inclusion, let Then,This implies thatSimilarly, . Again,Hence, .
Corollary 1. is the smallest -fuzzy congruence on containing .
4. L-Fuzzy Kernel Ideals
In this section, we study -fuzzy kernel ideals in Ockham algebras and give several characterizations for them.
Definition 11. An -fuzzy ideal on is called a kernel -fuzzy ideal if for some -fuzzy congruence of .
The following observation is immediate.
Lemma 9. If is an -fuzzy ideal of such that , then, for every ,from which it follows that
The following lemma gives an internal characterization for -fuzzy kernel ideals in Ockaham algebras.
Lemma 10. An -fuzzy ideal of is an -fuzzy kernel ideal if and only if it satisfies the following properties:(1)(2) for all
Proof. Let be an -fuzzy kernel ideal of . Then, there exists an -fuzzy congruence on such that . Let . If , then . Let , and consider the following:Hence, , for all . Therefore, (1) holds. To prove (2), let . If , clearly (2) holds. Suppose that . Then,Conversely, suppose that is an -fuzzy ideal of satisfying conditions (1) and (2). Consider the -fuzzy congruence given in Lemma 8. Now, we claim that . Let . Then,and hence, . To show the other inclusion,Hence, . Therefore, .
Corollary 2. If is a proper kernel L-fuzzy ideal of , thenfor some .
Proof. Suppose that is a proper kernel L-fuzzy ideal of . Then, , for some -fuzzy congruence of and , for some . Put . Now, we claim that . Let . Then, the claim clearly holds if . Suppose that . Then,Hence, the claim is true.
Lemma 11. Arbitrary intersection of a family of kernel -fuzzy ideals of is a kernel -fuzzy ideal.
We denote the set of all kernel -fuzzy ideals of by .
Theorem 2. The set of kernels -fuzzy ideal of ordered by set inclusion forms a complete lattice.
Consider the subset . Then, we have the following results.
Theorem 3. (1) is a complete sublattice of (2)If is a kernel -fuzzy ideal of , then
Proof. Let be a nonempty subset of . Then, , for all . Now, we claim that .
Let . Suppose that . Then,Now, suppose that . Then,Hence, the claim is true. Since is greatest element of , we have is a complete sublattice of .
It follows from Lemma 10.
Lemma 12. Let be an -fuzzy ideal of . Then, the -fuzzy subset of defined byis an L-fuzzy ideal of containing .
Proof. Suppose that be an -fuzzy ideal of . Now, sinceWe have . Let . Then,Again sinceSimilarly, we have , and hence, . Therefore, . Again, since, for any ,we have . Therefore, is an -fuzzy ideal of containing .
In the following theorem, we give a necessary and sufficient condition for an -fuzzy ideal of an Ockham algebra to be a kernel -fuzzy ideal.
Theorem 4. An -fuzzy ideal of is a kernel -fuzzy ideal of if and only if and .
Proof. Suppose that is a kernel -fuzzy ideal of . Then, by Lemmas 10 and 12, we have and . Now, since, for any ,we have . Hence, and .
Conversely, suppose that is an -fuzzy ideal of satisfying the given condition. Then, by Lemma 11, it is enough to show thatLet if ; clearly, (57) holds. Suppose that . Then,Hence, is a kernel L-fuzzy ideal of .
Theorem 5. Let be an -fuzzy ideal of . Then,(1)If , then (2)The mapping defined by is a closure on and
Proof. Let . Then, . Let . Then, if , obviously, . Suppose that . Then,