Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 8841670 | https://doi.org/10.1155/2020/8841670

Wondwosen Zemene Norahun, "-Fuzzy Filters in Distributive Lattices", Advances in Fuzzy Systems, vol. 2020, Article ID 8841670, 9 pages, 2020. https://doi.org/10.1155/2020/8841670

-Fuzzy Filters in Distributive Lattices

Academic Editor: Antonin Dvorák
Received24 Jul 2020
Accepted30 Nov 2020
Published29 Dec 2020

Abstract

In this paper, we introduce the concept of -fuzzy filters in distributive lattices. We study the special class of fuzzy filters called -fuzzy filters, which is isomorphic to the set of all fuzzy ideals of the lattice of coannihilators. We observe that every -fuzzy filter is the intersection of all prime -fuzzy filters containing it. We also topologize the set of all prime -fuzzy filters of a distributive lattice. Properties of the space are also studied. We show that there is a one-to-one correspondence between the class of -fuzzy filters and the lattice of all open sets in . It is proved that the space is a space.

1. Introduction

In 1970, Mandelker [1] introduced the concept of relative annihilators as a natural generalization of relative pseudo-complement and he characterized distributive lattices with the help of these annihilators. The concept of coannihilators and -filters in a distributive lattice with greatest element “” was introduced by Rao and Badawy [2] and they characterized -filters in terms of coannihilators. For a filter in , is an ideal in the set of all coannihilators, and conversely is a filter in when is any ideal in . A filter of is called a -filter if .

In 1965, Zadeh [3] mathematically formulated the fuzzy subset concept. He defined fuzzy subset of a nonempty set as a collection of objects with grade of membership in a continuum, with each object being assigned a value between 0 and 1 by a membership function. Fuzzy set theory was guided by the assumption that classical sets were not natural, appropriate, or useful notions in describing the real-life problems, because every object encountered in this real physical world carries some degree of fuzziness. A lot of work on fuzzy sets has come into being with many applications to various fields such as computer science, artificial intelligence, expert systems, control systems, decision-making, medical diagnosis, management science, operations research, pattern recognition, neural network, and others (see [47]).

In 1971, Rosenfeld used the notion of a fuzzy subset of a set to introduce the concept of a fuzzy subgroup of a group [8]. Rosenfeld’s paper inspired the development of fuzzy abstract algebra. Since then, several authors have developed interesting results on fuzzy subgroups (see [917]), fuzzy ideals of rings (see [16, 1821]), and fuzzy ideals of lattices (see [2228]).

Alaba and Norahun [29] studied the concept of -fuzzy ideals of a distributive lattice in terms of annulates. They also studied the space of prime -fuzzy ideals of a distributive lattice. In this paper, we introduce the dual of the concept of -fuzzy ideals which is called -fuzzy filters in a distributive lattice with greatest element “1.” We study the special class of fuzzy filters called -fuzzy filters. We prove that the set of all -fuzzy filters of a distributive lattice forms a complete distributive lattice isomorphic to the set of all fuzzy ideals of . We also show that there is a one-to-one correspondence between the class of prime -fuzzy filters of and the set of all prime ideals of . We prove that every -fuzzy filter is the intersection of all prime -fuzzy filters containing it. Moreover, we study the space of all prime -fuzzy filters in a distributive lattice. The set of prime -fuzzy filters of is denoted by . For a -fuzzy filter of , open subset of is of the form and is a closed set. We also show that the set of all open sets of the form forms a basis for the open sets of . The set of all -fuzzy filters of is isomorphic with the set of all open sets in .

2. Preliminaries

We refer to Birkhoff [30] for the elementary properties of lattices.

Definition 1 (see [2]). For any set of a lattice , define as follows:Here, is called the coannihilator of . If , we write instead of . Then, clearly and . For any subset of a lattice , it is clear that is a filter in .

Lemma 1. (see [2]). For any , the following conditions hold.(1)(2)(3)(4) if and only if The set of all coannihilator denotes . Each coannihilator is a coannihilator filter, and hence, for two coannihilators and , their supremum and infimum in arerespectively.

In a distributive lattice with 1, the set of all coannihilators of is a lattice and a sublattice of the Boolean algebra of coannihilator filters of .

For a filter in ,is an ideal in and the setis a filter of when is any ideal in . A filter of is called a -filter if .

Definition 2 (see [3]). Let be any nonempty set. A mapping is called a fuzzy subset of .
The unit interval together with the operations min and max form a complete lattice satisfying the infinite meet distributive law. We often write for minimum or infimum and for maximum or supremum. That is, for all , we have, and .
The characteristic function of any set is defined as

Definition 3 (see [8]). Let and be fuzzy subsets of a set . Define the fuzzy subsets and of as follows: for each ,Then, and are called the union and intersection of and , respectively.
For any collection, of fuzzy subsets of , where is a nonempty index set, and the least upper bound and the greatest lower bound of the ’s are given for each ,respectively.
For each , the setis called the level subset of at [3].

Definition 4 (see [27]). A fuzzy subset of a lattice is called a fuzzy ideal of if; for all , the following conditions are satisfied:(1)(2)(3)

Definition 5 (see [27]). A fuzzy subset of a lattice is called a fuzzy filter of if, for all , the following condition is satisfied:(1)(2)(3)In [27], Swamy and Raju observed the following:(1)A fuzzy subset of a lattice is a fuzzy ideal of if and only if(2)A fuzzy subset of a lattice is a fuzzy filter of if and only ifLet be a fuzzy subset of a lattice . The smallest fuzzy filter of containing is called a fuzzy filter of induced by and denoted by and

Lemma 2 (see [23]). For any two fuzzy subsets and of a distributive lattice , we have

The above result works dually.

For any two fuzzy subsets and of a distributive lattice , we have

The binary operations “” and “” on the set of all fuzzy subsets of a distributive lattice are as follows:If and are fuzzy ideals of , then and If and are fuzzy filters of , then and

The set of all fuzzy filters of is denoted by .

3. -Fuzzy Filters

In this section, we introduce the concept of -fuzzy filters in a distributive lattice with greatest element “1.” We study some basic properties of the class of -fuzzy filters. We prove that the class of -fuzzy filters forms a complete distributive lattice isomorphic to the class of fuzzy filters of . We also show that there is a one-to-one correspondence between the set of all prime -fuzzy filters of and prime fuzzy ideals of . Finally, we observe that every -fuzzy filter is the intersection of all prime -fuzzy filters containing it.

Throughout the rest of this paper, stands for the distributive lattice with greatest element “1” unless otherwise mentioned.

Theorem 1. Let be a fuzzy filter of . Then, the fuzzy subset of defined byis a fuzzy ideal of .

Proof. Let be a fuzzy filter of . Clearly, . For any ,Thus, .
On the other hand,Similarly, . So,Hence, is a fuzzy ideal of .

Lemma 3. Let be a fuzzy ideal of . Then, the fuzzy subset of defined as is a fuzzy filter of .

Proof. Let be a fuzzy ideal of . Since is the smallest element of , we get . For any ,Thus, is a fuzzy filter of .

Lemma 4. If and are fuzzy filters of , then implies .

Lemma 5. If are fuzzy ideals of , then implies .

Theorem 2. The set of all fuzzy ideals of forms a complete distributive lattice, where the infimum and supremum of any family of fuzzy ideals are given by

Theorem 3. The mapping is a homomorphism of into .

Proof. Let be two fuzzy filters of . Then, by Lemma 4, we have and . For any ,Since and , we get . Based on this, we haveThus, . So, .
On the other hand,Then, . Thus, . So, is a homomorphism.

Corollary 1. For any two fuzzy filters and of , we have

Proof. For any , . Since is a homomorphism, we have .

Lemma 6. For any fuzzy ideal of .

Proof. Since is a fuzzy ideal of , by Lemma 3, is a fuzzy filter of and is a fuzzy ideal of . Now, we proceed to show .Thus, .

Lemma 7. For any fuzzy filter of , the map is a closure operator on . That is,(1)(2)(3), for any two fuzzy filters of Now, we define -fuzzy filter.

Definition 6. A fuzzy filter of is called a -fuzzy filter of if .
Thus, -fuzzy filters are simply the closed elements with respect to the closure operator of Lemma 7, and is the smallest -fuzzy filter containing , for any fuzzy filter of .

Theorem 4. For a nonempty fuzzy subset of , is a -fuzzy filter if and only if each level subset of is a -filter of .

Proof. Let be a -fuzzy filter of . Then, . Now, we proceed to show for all . We know that . To show the other inclusion, let . Then, , and there is such that . Thus, . So, . Therefore, each level subset of is a -filter of .
Conversely, assume that each level subset of is a -filter. Then, is a fuzzy filter and . To prove our claim, let . Then, for each , there is such that and , which implies and . This shows that . Thus, . So, is a -fuzzy filter of .

Corollary 2. For a nonempty subset of , is a -filter if and only if is a -fuzzy filter of .

Theorem 5. Let be a fuzzy filter of . Then, is a -fuzzy filter if and only if, for each implies .

Proof. Let be a -fuzzy filter of and such that . Then,Conversely, suppose that for each implies . For any ,Thus, is a -fuzzy filter of .

Theorem 6. A fuzzy filter of is a -fuzzy filter if and only if

Proof. Suppose a fuzzy filter of is a -fuzzy filter. Then, by Theorem 4, every level subset is a -filter of . Let such that . Since is a -filter of , then , which implies for all . Thus, for each . So,Suppose conversely that the condition holds. To prove is a -fuzzy filter, it suffices to show that . For any , . If , then . By the assumption, for each . This shows that is an upper bound of . Thus, for all . So, . Hence, is a -fuzzy filter of .
Let us denote the set of all -fuzzy filters of by .

Theorem 7. The class of all -fuzzy filters of forms a complete distributive lattice with respect to set inclusion.

Proof. Clearly, is a partially ordered set. For , defineThen, clearly . We need to show is the least upper bound of . Since , is an upper bound of . Let be any upper bound for in . Then, , which implies that . Therefore, is the supremum of both in . Hence, is a lattice.
We now prove the distributivity. Let . Then,Thus, is a distributive lattice.
Now, we proceed to show the completeness. Since and are -filters, and are least and greatest elements of , respectively. Let . Then, is a fuzzy filter of and .Thus, . So, is a complete distributive lattice.

Theorem 8. The set is isomorphic to the lattice of fuzzy ideals of .

Proof. DefineLet and . Then, . Thus, . So, . Hence, is one to one.
Let . Then, by Lemma 3, is a fuzzy filter of . Now, we proceed to show that is a -fuzzy filter of . Let . Then, . Thus, by Lemma 6, we get that . So, . Thus, for each . Therefore, is onto.
Now, for any , . Similarly, . Therefore, is an isomorphism of onto the lattice of fuzzy filters of .

Theorem 9. The following are equivalent for each non-constant -fuzzy filter of .(1)For all ,(2)For any fuzzy points and of ,(3)For all ,

Proof. . Let such that . Then, . Since is a distributive lattice, by dual of Lemma 2, we have . Since and are fuzzy filters of , by the assumption, or . This shows that or .. Let such that . Now, we need to show or . Suppose not. Then, and , which implies there exist such that and . Put and . Then, and . Since , we have . By the assumption, we get that or , which is a contradiction. Thus, or .. Suppose such that . Then, by Corollary 1 we have . Since and are -fuzzy filters, by the assumption, we get that or , which implies or .

Definition 7. By a prime -fuzzy filter, we mean a nonconstant -fuzzy filter of satisfying (1) and hence all of the conditions of Theorem 9.
We have proved in Theorem 8 that there is an order isomorphism between the class of -fuzzy filters and the set of fuzzy ideals of . Now, we show that there is an isomorphism between the prime -fuzzy filters and the prime fuzzy ideals of the lattice of coannihilators.

Theorem 10. There is an isomorphism between the prime -fuzzy filters and the prime fuzzy ideals of the lattice of coannihilator.

Proof. By Theorem 8, the map is an isomorphism from into . Let be a prime -fuzzy filter of . Then, . Now, we prove is a prime fuzzy ideal of . Let such that . Since is onto, there exist such that and . Thus, . Since is an isotone, we have . Thus, . Since is a prime fuzzy filter, either or . This shows that either or . Thus, or . Hence, is a prime fuzzy ideal of .
Conversely, suppose that is a prime fuzzy ideal in . Since is onto, there exists a -fuzzy filter in such that . Let such that . Since is an isotone, we get . Thus, . Since is a prime fuzzy ideal of , either or . This implies or . Since is a -fuzzy filter, we get or . Thus, is prime fuzzy filter in . So, the prime -fuzzy filters correspond to prime fuzzy ideals of .

Theorem 11. Let be a -fuzzy filter of and be a fuzzy ideal of such that , . Then, there exists a prime -fuzzy filter of such that and .

Proof. Put . Since , is nonempty, and it forms a poset together with the inclusion ordering of fuzzy sets. Let be any chain in . Then, clearly is a -fuzzy filter. Since for each , we get that . Thus, . By applying Zorn’s lemma, we get a maximal element, say ; that is, is a -fuzzy filter of such that and .
Now, we proceed to show is a prime fuzzy filter. Assume that is not prime fuzzy filter. Let such that and . If we put and , then both and are -fuzzy filters of properly containing . Since is maximal in , we get . Thus, and . This implies there exist such that and , which implies . This shows that . This is a contradiction. Thus, is prime -fuzzy filter of .

Corollary 3. Let be a -fuzzy filter of , and . If , then there exists a prime -fuzzy filter of such that and .

Proof. Put . Since , is nonempty, and it forms a poset together with the inclusion ordering of fuzzy sets. Let be any chain in . Clearly, is a -fuzzy filter. Since for each , is an upper bound of . Thus, . So, is a -fuzzy filter containing and . Hence, . By applying Zorn’s lemma, we get a maximal element, say ; that is, is a -fuzzy filter of such that and .
Now, we proceed to show is a prime fuzzy filter. Assume that is not prime fuzzy filter. Let and and . If we put and , then both and are -fuzzy filters of properly containing . Since is maximal in , we get . Thus, and . Now, . This is a contradiction. Hence, is prime -fuzzy filter.

Corollary 4. Any -fuzzy filter of is the intersection of all prime -fuzzy filters containing it.

Proof. Let be a proper -fuzzy filter of . Consider the following.Clearly, . Assume that . Then, there is such that