Kernel L-Ideals and L-Congruence on a Subclass of Ockham Algebras
In this paper, we study -congruences and their kernel in a subclass of the variety of Ockham algebras . We prove that the class of kernel -ideals of an Ockham algebra forms a complete Heyting algebra. Moreover, for a given kernel -ideal on , we obtain the least and the largest -congruences on having as its kernel.
The concept of an Ockham algebra was first introduced by Berman , in 1977. Next, it has been studied by Urquhart et al. , Goldberg et al. [3, 4], and Blyth and Varlet . Blyth and Silva  presented the concept of kernel ideals in Ockham algebra. Wang et al.  presented Congruences and kernel ideals on a subclass of the variety of Ockham Algebras (in which ). 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  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 . He defined a fuzzy subset of a nonempty set as a function from to . Goguen in  presented the notion of -fuzzy subsets by replacing the interval with a complete lattice in the definition of fuzzy subsets. Swamy and Swamy  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  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 Kernel -ideals and -Congruence on a subclass of of the variety of Ockham algebras and study their characteristics. We prove that the class of kernel -ideal of -algebra forms Heyting algebras. Also, we get the least and the biggest -congruences, respectively, on -algebra having a given -ideal as an -kernel.
This section contains basic definitions and important results which will be used in the sequel.
Definition 1.. (see ). An algebra of type is said to be an Ockham algebra if(1) is a bounded distributive lattice(2) is a dual endomorphism
In this paper, for simplicity, any Ockham algebra is denoted by a pair . If the dual endomorphism on satisfies that , then this subclass is the Berman subclass .
Definition 2. (see ). An equivalence relation on is a congruence on if it is lattice congruence on and for every
Definition 3. (see ). An ideal of is said to be a kernel ideal if there is a congruence on such that
Definition 4. (see ). A Heyting algebra is an algebra of type where is a bounded distributive lattice and is a binary operation on such that for every , .
Lemma 1. (see ). Let and . If and , then(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
Throughout this article, is a none trivial complete lattice satisfying infinite meet distributive law: , , and . An -fuzzy subset of a nonempty is a mapping from into .
In this work, for simplicity, we say -subsets instead of -fuzzy subsets and write to say that is an -subset of .
The union and intersection of any class of -subsets of , respectively, represented by and , are defined as follows:for all , respectively.
Definition 5. (see ). For every and in , define a binary relation“” on by
It can be easily proved that is a partial ordering on the set of -subsets of and the poset forms a complete lattice in which for any ,
The partial ordering is called the point wise ordering.
For and , the setis called the -level subset of and for each , we have
For any we write to denote the constant -subset of which maps every element of onto .
Definition 6. (see ). Suppose is a function from into , and suppose is an -subset of and is an -subset of . Then, the image of under , , is an -subset of defined as for each ,The preimage of under , , is an -subset of and for each .
Definition 7. (see ). An -fuzzy subset of a lattice is called an -fuzzy ideals of if , and .
An -fuzzy subset of a lattice is called an -fuzzy filters of , if , and , for each .
It was also proved in  that an -subset of a lattice with is called an -ideal of if and for each . Dually, an -subset of a lattice with is called an -filter of if and , for each .
An -ideal (respectively, -filter) of is called proper if it is not a constant map .
Definition 8. (see ). An -subset of is said to be an -down set (respectively, -up set) if , then (respectively, ) for every .
Lemma 2. (see ). Let be an -subset of . Then, the -subset of defined byis the least -down set containing
Dually, we have the next result.
Lemma 3. (see ). Let be an -subset of . Then, the -fuzzy subset of defined byis the least -up set including
In what follows, for an Ockham algebra , we shall denote by the set of all kernel -ideals of and by the lattice of -ideals of in which the lattice operations and are given by
By an -binary relation on a nonempty set , we mean an -subset of . For an -binary relation on and each , the setis called the level binary relation of on .
Definition 9. (see ). An -relation on a nonempty set is said to be(1)Reflexive if , for all (2)Symmetric if , for each (3)Transitive if for each : for all An -equivalence relation on is a reflexive, symmetric, and transitive -relation on .
Definition 10. (see ). is an -equivalence relation on and is called an -congruence relation on if it compatible with and a unary operation .
For any and is -congruence relation, -subset of is defined as follows:
We call an -congruence class of determined by , and in particular, is called the kernel of and is called the cokernel of . One can easily observe that the kernel of is an -ideal of and the cokernel of is an -filter of .
Put and are binary operations and is a unary operation on expressed as follows:
After routine work, it can be proved that is an Ockham algebra and it is said to be the quotient Ockham algebra of modulo .
Definition 11. (see ). An -ideal of is called a kernel -ideal if for some -congruence of .
Lemma 4. (see ). An -ideal of is a kernel -ideal if and only if it holds the following conditions:(1)(2) for each
Lemma 5. (see ). The intersection of a class of kernel -ideals of is a kernel -ideal.
3. Kernel -Ideals in a Subclass of Ockham Algebra
In the present topic, we present the structure of the set of kernels -ideals in a subclass of the class O of Ockham algebra.
Lemma 6. Let . Then, any kernel -ideal of is determined by an L-filter of .
Proof. Suppose that be a kernel -ideal of . This implies that there exists an -congruence on such that . Put which is an -filter of .
Consider an -subset of defined as follows:To determine by the -filter of , we want to show that . Now, for any Since .
Now, we get thatTherefore, .That is, and hence . Thus, the result holds.
Next, we see an equivalent characterization of an -fuzzy ideal of a -algebra to get a kernel -ideal.
Lemma 7. Let . Then, an -ideal of is a kernel -ideal if and only if the following conditions are satisfied:(1), for each (2), for each
Proof. Assume that is a kernel -ideal of . Let . Then, by Lemma 4 (1), we have . Hence, (1) holds.
For each ,Thus, (2) is proved. Conversely, suppose that the given conditions hold. Let ,Hence, Again for every ,Then, it follows from Lemma 4 that is a kernel -ideal.
In the following result, we characterized the least kernel -ideal of .
Theorem 1. Let and be an -ideal of . Let be an -subset of defined bywhere and are as stated in Lemma 1. Then, is the least kernel -ideal of containing .
Proof. First we prove that is an -ideal of .Hence, .
Now, for each ,This implies that for each .
On the other side,With similarly approach, we can prove that and so . Therefore, and thus is an -ideal of .
Next, we prove that is a kernel -fuzzy ideal of .Thus, , for each , and the property (1) of Lemma 7 holds.
To see the property (2) of Lemma 7 holds, let Then,Hence, by Lemma 7, is a kernel -ideal of . Suppose is a kernel ideal of such that .
This implies there exists an -congruence on such that Let . As we clearly getAnd hence . Again for each Similarly, we can show that .
This implies thatAnd so . Therefore, from these observations, is the least kernel -ideal of including .
The following corollaries follows Theorem 1.
Corollary 1. Suppose and is an -ideal of . Then, is a kernel L-ideal if and only if .
Corollary 2. If , then the following properties hold, for all :(1)(2)(3)(4)
Given a - algebra and -ideals , of , we shall define
Suppose that and kernel -ideals of , then . Indeed, since and are kernel -ideals of , we have and , and is also a kernel -ideal. Thus,
Then, we have the following.
Theorem 2. Let and denote the set of all kernel -ideals. Then, is a complete bounded distributive lattice.
Proof. Suppose that . Then, is a kernel -ideal and the infimum of and . It follows by Theorem 1 that is a kernel -ideal and . Suppose that is a kernel -ideal of such that is an upper bounded of and , then and there follows . Hence, is the supremum of both and in . Thus, is a lattice. Obviously, and are the least and biggest kernel -ideals in , respectively. This implies that is bounded. The completeness is clear since the intersection of a family of kernel -ideals is also a kernel -ideal of (Lemma 4). As far as the distributivity, let . SinceIt follows that is a distributive lattice.
In order to further characterize the structure of the lattice of kernel -ideals of a -algebra, we require the following:
Theorem 3. Let . Then, is a Heyting algebra in which for , the relative pseudocomplement of and is defined as for every
Proof. Since andwe have .Similarly, we can prove that and henceOn the other side, for each ,Thus, , for each , and hence is an -ideal of .
Next, we show that is a kernel -ideal of .Also, for any ,Hence, by Lemma 4, is kernel -ideals of .
Next, we prove that is the biggest kernel -ideals of such that . Suppose that such that . Now, since for each We have clearlyAnd hence Therefore, is the pseudocomplement of relative to .
Theorem 4. Let and be a proper kernel -ideal of and . Then, is the intersection of a family of prime kernel -ideals including it.
Proof. Assume that is a proper -ideal of . Then, there exists such that Put and consider the setClearly, and so 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 in ; we prove that is a prime kernel -ideal. Let and be -ideals of such that . Suppose on contrary that and . Then,By the maximality of in , it follows that and . So that and Now,which is a contradiction.
Let be the set of prime kernel -ideals that contain . Then, . Suppose that . Then, there is such that This implies that and hence is proper. Thus, from the above, there exists a prime kernel -ideal which contains and and hence which is a contradictions. Hence, and so .
4. L-Congruences of a -Algebra
In this topic, we characterize an interesting property of the least -congruence on the -algebra such that the given kernel -ideal of as its -congruence class.
Let be a kernel -ideal of As shown in , the smallest -congruence on with kernel which is given by: for
Theorem 5. If and , then
Proof. Let . It is easily proved thatThen, we have . On the other side, let . Then, by (46), we haveConsider the following equation:By (49) and (50), we haveFrom (51), we haveAlso, from (51), we have