P-Fuzzy Ideals and P-Fuzzy Filters in P-Algebras

Norahun, Wondwosen Zemene

doi:https://doi.org/10.1155/2021/4561087

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 2021 | Article ID 4561087 | https://doi.org/10.1155/2021/4561087

P-Fuzzy Ideals and P-Fuzzy Filters in P-Algebras

Wondwosen Zemene Norahun¹

Academic Editor: Antonin Dvorák

Received28 Apr 2021

Accepted14 Jun 2021

Published01 Jul 2021

Abstract

In this paper, we introduce the concept of p-fuzzy ideals and p-fuzzy filters in a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Moreover, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filter.

1. Introduction

The concept of fuzzy sets was firstly introduced by Zadeh [1]. In 1971, Rosenfeld used the notion of a fuzzy subset of a set to introduce the concept of a fuzzy subgroup of a group [2]. Rosenfeld’s paper inspired the development of fuzzy abstract algebra. Since then, several authors have developed interesting results on fuzzy theory (see [3–14]). In this paper, we introduce the concept of p-fuzzy ideals and p-fuzzy filters in p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. Moreover, we prove that, for any fuzzy ideal of , there is the smallest p-fuzzy ideal containing it. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Moreover, we prove that the image and inverse image of a p-fuzzy ideal is a p-fuzzy ideal under a -epimorphism mapping. Finally, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filters.

2. Preliminaries

In this section, we recall some definitions and basic results on p-algebra and fuzzy theory.

Definition 1 (see [15]). An algebra of type is a p-algebra if the following conditions hold:(1) is a bounded lattice(2)For all

Theorem 1 (see [15]). For any two elements of a p-algebra, we have the following:(1)(2)(3)(4)(5)(6)(7)An element of a p-algebra is called closed, if .

Definition 2 (see [15]). A nonempty subset of is called an ideal of if for any and .

Definition 3 (see [15]). A nonempty subset of is called a filter of if for any and .

Definition 4 (see [16]). An ideal of is called a p-ideal if any , .

Definition 5 (see [16]). A filter of is called a p-filter if for any , .

Definition 6 (see [1]). Let be any nonempty set. A mapping is called a fuzzy subset of .
The unit interval together with the operations min and max forms a complete lattice satisfying the infinite meet distributive law; i.e.,for all and any .
We often write for minimum or infimum and for maximum or supremum. That is, for all , we have and .
The characteristics function of any set is defined as

Definition 7 (see [2]). Let and be fuzzy subsets of a set . Define the fuzzy subsets and of as follows: for each , and .
Then, and are called the union and intersection of and , respectively.
For any collection, of fuzzy subsets of , where is a nonempty index set, the least upper bound and the greatest lower bound of the ’s are given for each , and , respectively.
For each , the setis called the level subset of at [1].

Definition 8 (see [2]). Let be a function from into , be a fuzzy subset of , and be a fuzzy subset of .(1)The image of under , denoted by , is a fuzzy subset of defined by the following: for each , (2)The preimage of under , denoted by , is a fuzzy subset of defined by for each ,

Definition 9 (see [17]). A fuzzy subset of a bounded lattice is called a fuzzy ideal of , if for all the following conditions are satisfied:(1)(2)(3)

Definition 10 (see [17]). A fuzzy subset of a bounded lattice is called a fuzzy filter of , if for all the following conditions are satisfied:(1)(2)(3)We define the binary operations “” and “” on the set of all fuzzy subsets of asIf and are fuzzy ideals of , then and is a fuzzy ideal generated by .
If and are fuzzy filters of , then (the pointwise infimum of and ) and (the supremum of and ).

Theorem 2 (see [18]). Let be a lattice, and . Define a fuzzy subset of aswhich is a fuzzy ideal of .

Remark 1 (see [18]). is called the -level principal fuzzy ideal corresponding to .
Similarly, a fuzzy subset of defined asis the -level principal fuzzy filter corresponding to .

Remark 2. Let be a lattice with . Then, is called 0-distributive if for any with implying .
Throughout the rest of this paper, stands for the 0-distributive p-algebra unless otherwise mentioned.

3. P-Fuzzy Ideals

In this section, we study the concept of p-fuzzy ideals of a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. Moreover, we prove that, for any fuzzy ideal of , there is the smallest p-fuzzy ideal containing it. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. Finally, we show that the image and inverse image of a p-fuzzy ideal is a p-fuzzy ideal under a -epimorphism mapping.

Definition 11. A fuzzy ideal of is called a p-fuzzy ideal if for each .

Lemma 1. A fuzzy ideal of is a p-fuzzy ideal if and only if for all .

Lemma 2. For any , is a p-fuzzy ideal.

Proof. Let . If , then and . Thus, . If , then and . Thus, . Hence, is a p-fuzzy ideal.

Theorem 3. is a p-fuzzy ideal if and only if is a closed element.

Proof. Let be a p-fuzzy ideal. Then, and . Since , we have . Thus, is closed.
Conversely, suppose that is a closed element. Let . If , then and . Thus, . If , then and . Thus, . Hence is a p-fuzzy ideal.

Corollary 1. The following conditions on are equivalent:(1)Every fuzzy ideal is a p-fuzzy ideal(2)Every level principal fuzzy ideal is a p-fuzzy ideal(3) is Boolean algebra

Proof. The proofs of and are straightforward. To show that , suppose that every level principal fuzzy ideal is a p-fuzzy ideal. Then, by the above theorem, every element of is closed. For any , the supremum is given by .
To show that is distributive, it suffices to prove thatFor this purpose, let . Then, gives and . Similarly, and therefore . It follows from this that and hence that .
To see that is also complemented, observe that and . Since every , we have and . We see that the complement of is . Thus, is complemented. Hence is a Boolean algebra.

Theorem 4. A fuzzy subset of is a p-fuzzy ideal if and only if every level subset of is a p-ideal of .

Corollary 2. A nonempty subset of is a p-ideal if and only if is a p-fuzzy ideal.

In the following result, we prove that, for any fuzzy ideal of , there is the smallest p-fuzzy ideal containing it.

Theorem 5. Let be a fuzzy ideal of . Define

Then, is the smallest p-fuzzy ideal containing and hence is a p-fuzzy ideal if and only if .

Proof. Let be a fuzzy ideal of . Then clearly is a fuzzy ideal of . To prove is a p-fuzzy ideal, let . Clearly . On the other hand, . Thus, is a p-fuzzy ideal containing .
Now we proceed to show that is the smallest p-fuzzy ideal containing . Let be a p-fuzzy ideal containing . Let such that . Then, . Since is a p-fuzzy ideal and , we get that and . This shows that is an upper bound of . This implies . Thus, . So is the smallest p-fuzzy ideal containing .

Lemma 3. If and are fuzzy ideals of , then implies .

Lemma 4. For any two fuzzy ideals and of , .

Proof. Let and be two fuzzy ideals of . Clearly . To show the other inclusion, let . ThenThus, .

Lemma 5. For any fuzzy ideal of , the map is a closure operator on . That is,(1)(2)(3), for any two fuzzy ideals of p-fuzzy ideals are simply the closed elements of with respect to the closure operator.

Every p-fuzzy ideal is a fuzzy ideal but the converse may not be true. For this, we have the following example.

Example 1. Consider the p-algebra whose Hasse diagram is given below.
Define fuzzy subsets and of as follows:Then, it can be easily verified that and are p-fuzzy ideals of . Moreover, we observe that the fuzzy ideal of is not a p-fuzzy ideal.
The set of all p-fuzzy ideals of is denoted by . We now prove that is a lattice.

Theorem 6. If , then the supremum of and is given by

Proof. Put . Clearly . For any ,If and , then . ThusThus, .
On the other hand,Thus, .
To show is a p-fuzzy ideal, let . Then clearly .Thus, is a p-fuzzy ideal of .
Now we proceed to show that is the smallest p-fuzzy ideal containing and . Clearly and . Let be any p-fuzzy ideal containing and . For any ,This shows that . Thus, . So is the smallest p-fuzzy ideal of containing and .

Theorem 7. The set of p-fuzzy ideal of forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.

Proof. Clearly is a partially ordered set. For , the infimum and the supremum of and are and , respectively. Thus, is a lattice.
To show the distributivity, it suffices to show for all . For any ,If , then . Since is a p-algebra, for all , and we get that . ThusHence . So is distributive.
Now we show the completeness. Since is a poset and and are the greatest and least elements of , respectively, then is a complete distributive lattice.

Theorem 8. Let be a p-fuzzy ideal of and be a fuzzy filter of such that , . Then, there exists a prime p-fuzzy ideal 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 p-fuzzy ideal. Since for each , then . Thus, . By applying Zorn’s lemma, we get a maximal element. Let us say ; that is, is a p-fuzzy ideal of such that and .
Now we proceed to show is a prime fuzzy ideal. Assume that is not prime fuzzy ideal. Let and such that and . If we put and , then both and are p-fuzzy ideals of properly containing . Since is maximal in , we get and . Thus, and . This implies there exist such that and . So .This is a contradiction. Hence is prime p-fuzzy ideal of .
Let and be p-algebras. Then, a lattice morphism is said to be a -morphism if for all .

Theorem 9. Let be a -epimorphism. If is a p-fuzzy ideal of , then is a p-fuzzy ideal of .

Proof. Let be a p-fuzzy ideal of . Then, is a fuzzy ideal of . To show is a p-fuzzy ideal, let . Since , we have . On the other hand,Since and is a -morphism, we have . ThusThus, . So is a p-fuzzy ideal of .

Theorem 10. Let be a -epimorphism. If is a p-fuzzy ideal of , then is a p-fuzzy ideal of .

Theorem 11. Let be a -epimorphism. Then, the map defined by is a lattice epimorphism.

Proof. Let . Then, and are p-fuzzy ideals of . Thus, by Theorem 9, and are p-fuzzy ideals of . Clearly . For any ,If and , then . Thus,So .
Again clearly, . For any ,If and , then . Put and . Then, and . Based on this fact, we haveThus, . So is a homomorphism. Now we proceed to show is an epimorphism. Let . Then, by Theorem 10, . Since is onto, we have . Thus, is onto. So is a lattice epimorphism.

4. P-Fuzzy Filters

In this section, we study the concept of p-fuzzy filter of a p-algebra. We prove that the class of p-fuzzy filters forms a complete distributive lattice.

Definition 12. A fuzzy filter of is called a p-fuzzy filter of if for any .

Theorem 12. A fuzzy subset of is a p-fuzzy filter if and only if every level subset of is a p-filter of .

Corollary 3. A nonempty subset of is a p-filter if and only if is a p-fuzzy filter.
The set of all dense elements of is denoted by .

Corollary 4. is the smallest p-fuzzy filter of .

Proof. Since the set of all dense elements of is a p-filter of , by the above corollary is a p-fuzzy filter of . To show is the smallest p-fuzzy filter, let be a p-fuzzy filter of . Then clearly . If , then . Since is a p-fuzzy filter, we get that . This implies . Thus, is the smallest p-fuzzy filter.

Corollary 5. is p-fuzzy filter of if and only if .

Proof. Suppose is p-fuzzy filter of . Then, by the above corollary . Conversely, let . Since is a p-fuzzy filer, we get that is a p-fuzzy filter of .

Theorem 13. A proper fuzzy filter of is a p-fuzzy filter if and only if for any such that and .

The set of all p-fuzzy filters of a p-algebra is denoted by . The following result shows that forms a lattice.

Theorem 14. If , then the supremum of and is given by

Proof. Put . Clearly . For any ,If and , then . ThusThus, .
On the other hand,Thus, .
To show is a p-fuzzy filter of , let .Thus, is a p-fuzzy filter of . We now show that is the smallest p-fuzzy filter containing and . Let be a p-fuzzy filter containing and . For any ,Thus, is the smallest p-fuzzy filter containing and .

Theorem 15. The set of p-fuzzy filter of forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.

Proof. Clearly is a partially ordered set. For , the infimum and the supremum of and are and , respectively. Thus, is a lattice.
To show the distributivity, it suffices to show for all . For any ,If , then . We can easily verify that . Thus,Thus, . So is distributive.

5. Relation between and

In this section, we show that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filters.

Lemma 6. Let be a p-fuzzy ideal. DefineThen, is a p-fuzzy filter.

Proof. Let be a p-fuzzy ideal of . Since , we get that . For any , . Thus, is a fuzzy filter. To show is a p-fuzzy ideal, let . Then, . Hence is a p-fuzzy filter of .

Lemma 7. Let be a fuzzy filter. Define

Then, is a p-fuzzy ideal.

Proof. Let be a fuzzy filter. Since , we get . For any ,Thus, is a fuzzy ideal of . To show is a p-fuzzy ideal, let . Then, . Thus, is a p-fuzzy ideal of .

Theorem 16. Let be a fuzzy filter of . Then, if and only if is a p-fuzzy filter.

Proof. Let be a fuzzy filter of and . Then, by Lemmas 6 and 7, is a p-fuzzy filter. Conversely, let be a p-fuzzy filter and . Then, . Thus, .

Theorem 17. In , .

Proof. Define by . Since , by Lemma 6, . Let such that . Then, . Since and are p-fuzzy filters of , then and are p-fuzzy ideals of and . To show , let . Then, . This implies . Since and are p-fuzzy ideal, we get that . Thus, is one to one.
Let . Then, by Theorem 16, . Since is a fuzzy filter, then is a p-fuzzy ideal of and . So is onto.
Let and . ThenThus, . On the other hand,Thus, . Similarly, . Hence .

6. Conclusion

In this paper, we introduced the concept of p-fuzzy ideals and p-fuzzy filters in a p-algebra. We provide a set of equivalent conditions for a fuzzy ideal to be a p-fuzzy ideal and a p-algebra to be a Boolean algebra. It is proved that the class of p-fuzzy ideals forms a complete distributive lattice. We also studied the image and inverse image of p-fuzzy ideals under a -epimorphism mapping. Moreover, we prove that there is an isomorphism between the class of p-fuzzy ideals and p-fuzzy filter. Our future work will focus on -fuzzy ideals in a (0, 1) distributive lattice.

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 study. The author is one of the academic staff of University of Gondar, College of Natural and Computational Sciences, Department of Mathematics, Gondar. Ethiopia.

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Copyright

Copyright © 2021 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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