<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 9.691 12.3952" width="9.691pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-234" d="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 422L401 276C355 192 240 56 188 56C163 56 154 89 163 133C184 233 207 338 235 443L230 448L152 424L58 17C40 -60 23 -143 23 -185C23 -241 42 -261 63 -261C92 -261 117 -241 125 -227L124 -221C109 -209 89 -165 89 -103C89 -55 96 -13 105 26H107C121 -3 134 -12 151 -12C172 -12 194 -5 221 16C279 61 330 124 384 194H386C381 159 377 141 368 105C343 3 363 -12 383 -12C416 -12 486 35 538 96Z"/></g></svg>-</span>Fuzzy Filters in Distributive Lattices

Norahun, Wondwosen Zemene

doi:https://doi.org/10.1155/2020/8841670

Advances in Fuzzy Systems

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

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

-Fuzzy Filters in Distributive Lattices

Wondwosen Zemene Norahun¹

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 [4–7]).

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 [9–17]), fuzzy ideals of rings (see [16, 18–21]), and fuzzy ideals of lattices (see [22–28]).

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 . Let . Consider the setBy the above corollary, we can find a prime -fuzzy filter of such that and . This implies . This shows that , which is a contradiction. Thus, . So, .

4. The Space of Prime -Fuzzy Filters

In this section, we study the space of prime -fuzzy filters of a distributive lattice and some properties of the space also.

Let be the set of all prime -fuzzy filters of a distributive lattice. Let , where is a fuzzy subset of and . We let , i.e., .

Lemma 8. For any fuzzy filters and of , we have(1)(2)(3)

Proof. (1)Let and . Then, and . Thus, .(2)Since , by (1), we have . Now, we proceed to show the other inclusion; let . Then, . This shows that either or . So, . Hence, .(3)By (1), we have . On the other hand, let . Then, and . Since is a prime fuzzy filter, we get that . This shows that . Thus, .

Lemma 9. Let be a fuzzy subset of . Then, .

Proof. To prove our claim, it suffices to show . Let . Then, . We need to show . Suppose not. Then, , which implies that , which is a contradiction. Thus, . So, .

Theorem 12. Let and . Then,(1)(2)(3)

Proof. (1)If , then either or . This shows that or . Thus, . So, . Hence, . On the other hand, let . Then, . This implies either or . Thus, .(2)If , then and . This implies and . This shows that . Since is prime fuzzy filter, card Im and is prime. Thus, , which implies . Thus, and hence . Conversely, let . Then, , which implies . Thus, and . This shows that and . Thus, . So, . Therefore, .(3)Clearly, . Let . Then, . This implies there is such that . If we take some such that , then . Thus, . So, .

Lemma 10. Let ; ; and . Then,

Proof. Let . Then, and . This implies that and . Since is prime filter and , we have and . This shows that . Thus, . So, . To show the other inclusion, let . Then, . This implies and . Thus, and . So, . Hence, .

Lemma 11. Let be any family of fuzzy filters of . Then,

Proof. Since for each , we have for each . Thus, .
Conversely, let . Then, for each . This implies . Thus, for any , is an upper bound of . This implies that . This shows that and . So, . Thus, . Hence, .

Theorem 13. The collection .

Proof. First, we define two fuzzy subsets of as follows: and for all . Then, and are fuzzy filters of . Since , for all , we get that . This shows that . Since each is nonconstant, for all . So . Hence, .
Next, let . Then, by Lemma 8, we get that . This shows that is closed under finite intersection.
Now, we proceed to show that is closed under arbitrary union. Let be any family of fuzzy filters of . Then, by Lemma 11 we havewhich implies . Thus, by Lemma 9, we get thatSo, is closed under arbitrary union. Therefore, is a topology on . The space will be called the space of prime -fuzzy filters in .
In the above theorem, we proved that the family of is a topology on . In the following result, we show that the set of all open sets of the form is a basis for the topology on .

Theorem 14. The collection forms base for some topology .

Proof. Let be any open set in and . Then, and there is such that . Put ; then and . To show , let . Then, and . This shows that . Thus, . Hence, for any open set in we can find in such that . Therefore, is a base for .

Theorem 15. The space is a -space.

Proof. Take any two different elements and in . Then, either or . Without loss of generality, we can assume that . Then, and . Thus, is a -space.

Theorem 16. For any fuzzy filter of , .

Proof. For any fuzzy filter of , we have and . Now we proceed to show the other inclusion; let . Then, . Suppose ; then . This implies , which is impossible. Thus, and hence .
In the following result, we show that there is a one-to-one correspondence between the class of -fuzzy filters and the lattice of all open sets in .

Theorem 17. The lattice is isomorphic with the lattice of all open sets in .

Proof. The lattice of all open sets in is . Define the mappingSince and is a -fuzzy filter, every open subset of is of the form for some . This shows that the map is onto.
Let . Now, we need to show . Suppose not. Then, , which implies that there is such that either or . Without loss of generality, we can assume that . Put . Then by Corollary 3, we can find a prime -fuzzy filter such that and . Thus, and . So, and . This is a contradiction. Hence, .
Now, we show that is homomorphism. Let . Then,Similarly, . This shows that is a homomorphism. Hence, is an isomorphism.
For any fuzzy subset of , is an open set of and is a closed set of . In the following result, we prove the closure of a fuzzy set.

Theorem 18. For any family , closure of is given by .

Proof. We know that closure of is the smallest closed set containing . To prove our claim, it is enough to show that is the smallest closed set containing . Since the set of all -fuzzy filter is a complete distributive lattice, is a -fuzzy filter and is a closed set in . If ; then . Thus, . This implies that . Let be any closed set in containing . Then, , for each . Thus, and . So, is the smallest closed set containing . Hence, .

5. Conclusion

In this work, we studied the concept of -fuzzy filters of a distributive lattice. We proved 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 observed that every -fuzzy filter is the intersection of all -fuzzy filters containing it. We also studied the space of all prime -fuzzy filters in a distributive lattice. Our future work will focus on -fuzzy ideals of a C-algebra.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2020 Wondwosen Zemene Norahun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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