Generalized <i>m</i>-Polar Fuzzy Positive Implicative Ideals of BCK-Algebras

Al-Masarwah, Anas; Ahmad, Abd Ghafur; Muhiuddin, G.; Al-Kadi, D.

doi:https://doi.org/10.1155/2021/6610009

Journal of Mathematics

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

Research Article | Open Access

Volume 2021 | Article ID 6610009 | https://doi.org/10.1155/2021/6610009

Generalized m-Polar Fuzzy Positive Implicative Ideals of BCK-Algebras

Anas Al-Masarwah,¹Abd Ghafur Ahmad,¹G. Muhiuddin,²and D. Al-Kadi³

Academic Editor: Feng Feng

Received23 Nov 2020

Revised26 Jan 2021

Accepted02 Feb 2021

Published17 Feb 2021

Abstract

This study focuses on combining the theories of -polar fuzzy sets over -algebras and establishing a new framework of -polar fuzzy -algebras. In this paper, we define the idea of -polar fuzzy positive implicative ideals in -algebras and investigate some related properties. Then, we introduce the concepts of -polar -fuzzy positive implicative ideals and -polar -fuzzy positive implicative ideals in -algebras as a generalization of -polar fuzzy positive implicative ideals. Several properties, examples, and characterization theorems of these concepts are considered.

1. Introduction

The inception of the idea of -algebras, presented by Imai and Iséki [1], laid the frameworks and foundations as well as gave birth to great research studies. Such algebras generalize Boolean D-poset (-algebras) as well as Boolean rings. -algebras have many applications in several fields, such as groups, semigroups, graphs, topology and functional analysis, and so on. The study of ideals forms an essential aspect of the theory of -algebra. Since Imai and Iséki [1] introduced the notion of ideals in -algebra, several kinds of ideals in -algebras have occurred, for example, H-ideals, positive implicative ideals, implicative ideals, and so on.

The essential idea of a fuzzy set, proposed by Zadeh [2] in 1965, provides a natural framework for generalizing many fundamental concepts of algebras. Moreover, the idea of fuzzy sets in /-algebras was proposed by Xi [3]. The theory of fuzzy algebraic structures plays a prominent role in different domains of mathematics and other sciences such as theoretical physics, topological spaces, real analysis, coding theory, set theory, logic, and information sciences. In 1994, bipolar fuzzy (BF) set theory was proposed by Zhang [4] as a new platform that extends crisp (classical) and fuzzy sets. BF sets’ membership grades (positive and negative) belong to the interval [−1, 1]. Hybrid models of fuzzy sets have been applied in various algebraic structures, for instance, hemirings [5], -algebras [6], and /-algebras [7]. In many real-world problems, multipolar information play a fundamental role in distinct areas of sciences, such as neurobiology and technology. Data sometimes come from components (); for example, consider the following statement “Harvard University is a Good University.” In this statement, the degree of membership may not be a real number in the standard interval . In fact, Harvard University is a good university in several components: good in ranking, location, facilities and education, etc. Any component may be a real number in . If we have components under consideration, then the degree of the membership of the fuzzy statement is an element of , that is, an -tuple of real number in . Based on the above discussion, Chen et al. [8] generalized the theory of BF set theory to get a new notion, called -polar fuzzy (-pF) set theory in 2014. In -pF sets, the grade of membership function is extended from [0, 1] to -power of or .

The framework of the fuzzy subgroup, initially presented by Rosenfeld [9] in 1971, is a fundamental concept of fuzzy algebras. Pu and Liu [10] and Murali [11] defined “quasi-coincidence” and “belongingness” of a fuzzy point with a fuzzy set, respectively. These notions played a fundamental role to establish distinct kinds of fuzzy subgroups. In the literature, Bhakat and Das [12] first generalized fuzzy subgroups to -fuzzy subgroups and they proposed and discussed the idea of -fuzzy subgroups. In this aspect, Dudek et al. [13], Ibrara et al. [14], Jun and Song [15], and Narayanan and Manikantan [16] extended these results to semigroups, near-rings, and hemirings. In /-algebras, Jun [17] presented -fuzzy ideals as an extension of fuzzy ideals. In [18], Zulfiqar introduced -fuzzy positive implicative ideals in -algebras. Zhan and Jun [19] defined -fuzzy ideals in -algebras.

The notion of -pF set theory was applied to many practical problems, particularly in the field of graph theory (see, for e.g., [20–22]). In [23], Sarwar and Akram applied -pF set theory to matroid theory. In addition, various applications of -pF sets and other hybrid models of fuzzy sets in pure and applied mathematics are studied in [24–32]. Recently, -pF set theory has been applied to various algebraic structures on different aspects, namely, Farooq et al. applied -pF set theory to groups [33], Akram and Farooq applied -pF set theory to Lie algebras [34, 35], and the authors applied -pF set theory to /-algebras (see [36–40]. Motivated by a lot of work on -pF sets, we present -polar fuzzy positive implicative (-pFPI) ideals in -algebras and discuss some related results. Using the concept of quasi-coincidence of an -pF point within an -pF set and as a generalization of -pFPI ideals, we introduce the concepts of -polar -fuzzy positive implicative () ideals and -polar -fuzzy positive implicative () ideals in -algebras. Several properties, examples, and characterization theorems of these concepts are considered.

2. Preliminaries

Here are some of the important concepts of -algebras, -pF sets, and -pF ideals that are useful for further discussions. Throughout this paper, for the convenience, stands for a -algebra.

An algebra of type (2, 0) is called a -algebra if the axioms below are satisfied for all :(i)(ii)(iii)(iv)(v) and imply where can be presented by if and only if . Every -algebra satisfies the following axioms for all :(1).(2).

Definition 1 (see [8]). An -pF set on is a mapping . The membership grade of any element is given aswhere is the -th projection mapping. The grades and are the largest and the smallest grades in , respectively.

Definition 2 (see [36]). An -pF set of is called an -pF ideal of if for any ,(1).(2).That is,(1)(2)for all .

Definition 3 (see [36]). The set , where is an -pF set of , is called be the level cut subset of for all .

Lemma 1 (see [36]). Every -pF ideal of satisfies the following assertion for all :

3. -Polar Fuzzy Positive Implicative Ideals

Definition 4. (see [41]). A nonempty subset of is called a positive implicative ideal of if for all ,(1).(2) and imply .

Definition 5. An -pF set of is called an -pFPI ideal of if for any ,(1).(2).That is,(1)(2)for all .

Example 1. Let and a binary operation “” be given as follows:Then, is a -algebra. Let be a 4-pF set defined asSince conditions (1) and (2) of Definition 5 are satisfied. Then, is a 4-pFPI ideal of .

Theorem 1. Any -pFPI ideal of is an -pF ideal of , but the converse does not hold.

Proof. Let be an -pFPI ideal of . Then, . By taking in Definition 5 (2) and since , we have . Hence, is an -pF ideal of .
The last part is shown by the following example.

Example 2. Let and a binary operation “” be given as follows:Then, is a -algebra. Let be a 3-pF set defined asSince conditions (1) and (2) of Definition 2 are satisfied. Then, is a 3-pF ideal of , but it is not a 3-pFPI ideal of since .
We now give the conditions for an -pF ideal to be an -pFPI ideal of .

Theorem 2. An -pF set of is an -pFPI ideal of if and only if it is an -pF ideal of and for all .

Proof. Suppose is an -pFPI ideal of . By Theorem 1, is an -pF ideal of . If is replaced by in Definition 5 (2), thenfor all .
Conversely, let be an -pF ideal of . Then, for all . Also, since for all , it follows by Lemma 1 thatNow, by assumptionHence, is an -pFPI ideal of .

Theorem 3. An -pF set of is an -pFPI ideal of if and only if is a positive implicative ideal of for all .

Proof. Suppose that is an -pFPI ideal of . Let be such that . Then, , and we have . Let be such that . Then, and . It follows from Definition 5 (2) thatThus, . Hence, is a positive implicative ideal of .
Conversely, assume that is a positive implicative ideal of for all . If there exists such that , then for some . Then, , a contradiction. Thus, for all . If there exist such that , thenfor some . It follows that and , but . This is a contradiction. Thus, for all . Hence, is an -pFPI ideal of .

4. -Polar -Fuzzy Positive Implicative Ideals

Al-Masarwah and Ahmad [40] extended the concepts of “belongingness” and “quasi-coincidence” of a fuzzy point with a fuzzy set and proposed the concepts of “belongingness” and “quasi-coincidence” of an -pF point with an -pF set as follows.

An -pF set of of the formis said to be an -pF point, written as , with support and value .

An -pF point (1)Belongs to , written as , if , i.e., , .(2)Is quasi-coincident with , written as , if , i.e., , .

We say that(1) if does not hold.(2) (resp., ) if or (resp., and ).

Next, we introduce ideals of and discuss several results.

Definition 6. An -pF set of a -algebra is called an ideal of if for all and ,(1).(2) and .

Example 3. Consider a -algebra which is given in Example 2. Let be an -pF set defined asThen, is an ideal of .

Theorem 4. An -pF set of is an ideal of if and only if for all ,(i).(ii).

Proof. Suppose that is an ideal of . Let and assume . If , thenfor some . This implies that , but . Since , we get . Therefore, , a contradiction. Thus, for all . If , then , and so . Hence, . Otherwise, , a contradiction. Hence, for all . Let . Assume thatThen,If not, thenfor some . This implies that and , but , a contradiction. Hence, whenever . If , thenIt follows that . Therefore, or . If , thena contradiction. Therefore, for all .
Conversely, suppose that (i) and (ii) hold. Let and be such that . Then, . Assume . If , thena contradiction. Therefore, , which implies thatThus, . Let and be such that and . Then,Suppose that . If , thena contradiction. Hence, . This implies thatSo, . Hence, is an ideal of .

Theorem 5. Every ideal of is an ideal of , but the converse does not hold.

Proof. Let be an ideal of . Then, (1) of Definition 6 holds. Put in Definition 6 (2), and we getSo,Hence, is an ideal of .
The last part is shown by the following example.

Example 4. Reconsider the -algebras given in Example 2. Let be a 3-pF set defined asThen, is a 3-polar -fuzzy ideal of , but it is not a ideal of since .

Lemma 2 (see [40]). Every ideal of satisfies the following assertion: for all ,

Theorem 6. Let be an ideal of . Then, is an ideal of if and only if for all .

Proof. Assume is an ideal of . If is replaced by in Theorem 4 (2), thenfor all .
Conversely, let be an ideal of . Then, for all . Also, since for all , it follows by Lemma 2 thatNow, by assumptionHence, is an ideal of .

Theorem 7. An -pF set of is an ideal of a -algebra if and only if is a positive implicative ideal of for all .

Proof. Assume that is an ideal of a -algebra . Let and . Then, . Theorem 4 (i) implies thatThus, . Again, let . Then, and . Theorem 4 (ii) implies thatTherefore, . Thus, is a positive implicative ideal of .
Conversely, let be an -pF set of be such that is a positive implicative ideal of for all . If there exists such that , thenfor some . It follows that , but , a contradiction. Therefore, for all . Suppose there exist such thatThen, for some . This implies that and , but . This is impossible. Thus,for all . Hence, is an ideal of by Theorem 4.

5. -Polar -Fuzzy Positive Implicative Ideals

In the current section, we introduce ideals of and discuss some relevant results and properties.

Definition 7 (see [38]). An -pF set of is called an ideal of if for all and ,(1).(2) implies or .

Theorem 8. (see [38]). An -pF set of is an ideal of if and only if for all ,(i).(ii).

Lemma 3 (see [38]). Any ideal of satisfies the following assertion for all :

Definition 8. An -pF set of is called an ideal of if for all and ,(1).(2) implies or .

Example 5. Consider a -algebra which is given in Example 2. Let be a 3-pF set defined asThen, is a ideal of .

Theorem 9. An -pF set of is an ideal of if and only if for all ,(i).(ii).

Proof. Let be an ideal of . Assume there exists such that . Then,By Definition 8 (1), we have , i.e., or . Since , . This is a contradiction. Hence, for all . Suppose there exist such that . Then,It follows that or . Then, or . Since and , it follows that , a contradiction. Hence, for all .
Conversely, let . Then, , either or . If , then , and so . That is, . If , then by (i), . We consider two cases: Case (1). If , then , and so . Case (2). If , then ; it follows that , and so .Again, let for . Then, . We consider two cases: Case (1). If , then Consequently, or . That is, or . Hence, or . Case (2). If , thenAssume or . Then, or . Thus, or . It follows that or , and so or . Hence, implies or .

Theorem 10. Every ideal of is an -polar -fuzzy ideal of , but the converse does not hold.

Proof. Let be an ideal of . Then, for all and , we getPut in (44), and we getThis impliesHence, is an -polar -fuzzy ideal of .
The last part is shown by the following example.

Example 6. Reconsider the -algebras given in Example 2. Let be a 3-pF set defined asThen, is an ideal of . But it is not an ideal of since .

Theorem 11. Let be an -polar -fuzzy ideal of . Then, is an ideal of if and only if for all .

Proof. Let be an ideal of . If is replaced by in Theorem 9 (ii), we getfor all .
Conversely, Assume that is an -polar -fuzzy ideal of and for all . Then, for all . Also, since for all , it follows by Lemma 3 thatNow, by assumption,Hence, is an ideal of .

Theorem 12. The intersection of any family of an ideal of is an ideal of .

Proof. Let be a family of ideals of and . Then,for all . Thus,Therefore, . Let . Since every is an ideal of ,for all . Thus,Therefore, . Hence, is an ideal of .

Theorem 13. An -pF set of is an ideal of if and only if is a positive implicative ideal of for all .

Proof. Assume that is an ideal of and . Suppose . Then, . Now,Thus, . Hence, . Let . Then, and . Now,Thus, , that is, . Therefore, is a positive implicative ideal of .
Conversely, assume is a positive implicative ideal of . Let be such that . Choose such thatThen, and . Since is a positive implicative ideal of , we have , and so , a contradiction. Hence, for all . Assume such that . Choose such thatThen, . Since and is a positive implicative ideal of , . That is, . This is a contradiction. Thus, for all . Hence, is an ideal of .

Corollary 1. Every -pFPI ideal of is an ideal of .

6. Conclusions

-pF algebraic structures play a significant role in several fields of pure and applied mathematics. In order to broaden -pF algebraic structures, in this study, we have introduced -pFPI ideals, ideals, and ideals in -algebras and discussed several relevant properties. We have considered some characterizations of these concepts in -algebras. In our further research, we will focus on adopting this approach to some more algebraic structures, such as -algebras, -algebras, semigroups, -semigroups, and hemirings, and to some more complicated applications from the domains of computer sciences and information systems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This study was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

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