Journal of Function Spaces

Journal of Function Spaces / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 6615288 | https://doi.org/10.1155/2021/6615288

M. Mursaleen, M. Balamurugan, K. Loganathan, Kottakkaran Sooppy Nisar, "()-Bipolar Fuzzy -Ideals of BCK/BCI-Algebras", Journal of Function Spaces, vol. 2021, Article ID 6615288, 8 pages, 2021. https://doi.org/10.1155/2021/6615288

()-Bipolar Fuzzy -Ideals of BCK/BCI-Algebras

Academic Editor: Alberto Fiorenza
Received16 Nov 2020
Revised23 Dec 2020
Accepted10 Jan 2021
Published11 Feb 2021

Abstract

In this paper, the idea of -bipolar fuzzy -ideals and an -bipolar fuzzy ideals of -algebras is delivered, and their related properties are investigated with the aid of some examples. We also provide the connection between -bipolar fuzzy ideals and bipolar fuzzy ideals and -bipolar fuzzy -ideals and bipolar fuzzy -ideals by way of counterexamples.

1. Introduction

In the real world, there are several difficult problems in engineering, medical science, economics, environment, social science, and other various fields involving incalculable data. Problems of this nature which are frequently encountered in our daily lives cannot be solved by classical mathematical methods. In 1965, the belief of fuzzy sets turned into was first delivered with the aid of Zadeh [1], which is an effective hand set for modelling indecision and elusiveness in numerous issues springing up inside the field of technology. For the ultimate four decades, the fuzzy idea has ended up as a very lively area of study, and plenty of developments have been made within the concept of fuzzy sets to find the fuzzy analogues of the classical set theory. The study of -algebras was initiated by Imai and Iséki [2, 3] in 1966 as showed in the concept of set-theoretic difference and propositional calculi. After that, many researchers investigated the fuzzification of ideals and subalgebras in -algebras.

Bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is [1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [1, 0) of an element indicates that the element somewhat satisfies the implicit counterproperty. The idea which lies behind such description is connected with the existence of “bipolar information” (e.g., positive information and negative information) about the given set. Positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. Actually, a wide variety of human decision-making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side. For instance, cooperation and competition, friendship and hostility, common interests and conflict of interests, effect and side effect, likelihood and unlikelihood, and feedforward and feedback are often the two sides in decision and coordination. In traditional Chinese medicine, “yin” and “yang” are the two sides. Yin is the feminine or negative side of a system, and yang is the masculine or positive side of a system. The coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical health of a person as well as for the stability and prosperity of a social system. Thus, bipolar fuzzy sets indeed have potential impacts on many fields, including artificial intelligence, computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social science. In 1998, the notion of bipolar fuzzy sets was proposed by Zhang [5, 6] as a generalization of fuzzy sets [1].

Bipolar-valued fuzzy units, which can be added by means of Lee [7], are an augmentation of fuzzy sets whose participation degree extend is broadened from the stretch [0, 1] to . Lee [8] involved the research with interval-valued fuzzy units, intuitionistic fuzzy units, and bipolar-valued fuzzy sets. Bipolar fuzzy sets have various applications in fuzzy algebras. For example, bipolar fuzzy ideals [4] in -semigroups, bipolar fuzzy subalgebras and ideals [7] of -algebras, bipolar fuzzy -ideals in -algebras, and bipolar valued fuzzy -algebras [9] are some of them. Bhakat and Das [10, 11] utilized the idea of -fuzzy subgroups by the usage of and among a fuzzy factor and a fuzzy subset. It is visible that -fuzzy subgroups are a critical generalization of Rosenfeld’s [12] fuzzy subgroup. Similar types of -fuzzy ideals of BCI-algebras are introduced by Zhan et al. [5]. Zhan et al. [13, 14] introduced three-way multiattribute decision-making based on outranking relations and multicriteria decision-making method based on a fuzzy rough set model with fuzzy -neighborhoods. Senapati et al. [15] introduced cubic intuitionistic implicative ideals of -algebras.

Jana et al. [16, 17] introduced generalizations of -intuitionistic fuzzy subalgebras and ideals of BCK/BCI-algebras with thresholds. Jana et al. [18] introduced the concept of -bipolar fuzzy -subalgebras and -bipolar fuzzy ideals of -algebras. These works are enough to motivate us, and, to the best of our knowledge, no other works are available on -bipolar fuzzy ideals and -ideals in -algebras and other fuzzy algebraic structures. For this reason, we have developed the theoretical study of -bipolar fuzzy ideals of -algebras and -bipolar fuzzy -ideals of -algebras. In this paper, the concepts of -bipolar fuzzy ideals are presented, and properties are established. Moreover, -bipolar fuzzy -ideals of -algebras are proposed, and their properties are examined in detail.

2. Preliminaries

Definition 1. An algebra of kind could be a -algebra if it fulfills for all ,
,
,
,
,
and .
We remind the reader that signifies a -algebra unless otherwise specified.
A nonempty subset of is called an ideal of if it satisfies
(I1) ,
(I2) .
A nonempty subset of is called a -ideal of if it satisfies and
(I3) .
A bipolar fuzzy set () in is denoted by , where are the maps from to and from to , respectively.

3. -Bipolar Fuzzy Ideals

In this section, we investigate an -bipolar fuzzy ideals of -algebras.

Definition 2 (see [18]). A is a of if it satisfies the subsequent assertions: (i)(ii)(iii)

Definition 3. Let be a in of the form A bipolar fuzzy point with help and values and is signified by . In a bipolar fuzzy point and a in a set , we offer significance to the symbol , where .
To say that and means that and , and in this case, we say that and are said to belong to (respectively, be quasicoincident with) a .
To say that and and imply or and or .
To say that imply does not hold and does not hold, where .

Definition 4. A is an of if it satisfies the subsequent assertions: (i), for all and (ii), for all and

Example 5. Consider a -algebra with the subsequent Cayley table: Define a of as follows: Hence, is an of .

Theorem 6. A is an of if and only if it satisfiesif and only if it satisfies for all .

Proof. Let be an of and . If and , then and .
Assume that and . Let us take and such that and . Then, and but and , a contradiction.
Hence, whenever and whenever .
Suppose that and . Then, , and , , which imply that Thus, and . Otherwise, and , a contradiction. And so, for all .
On the contrary, assume that of is valid. Let and and such that and . Then, and .
If and , then and .
Otherwise, we get a contradiction. In that case, Hence, and .
Therefore, is an of .

Lemma 7. Every is an of .
The opposite of Lemma 7 is not correct in general, justified in the subsequent Example 8.

Example 8. Consider a -algebra with Cayley table: Define a of as follows: Hence, of but is not of because .

4. -Bipolar Fuzzy -Ideals

In this section, we investigate an -bipolar fuzzy -ideals of -algebras.

Definition 9. A is a BFBI of if it satisfies Definition 2 (i) and the subsequent assertions: (i)(ii)

Definition 10. A is called an -BFBI of if it satisfies the subsequent assertions: (i), for all and (ii), for all and

Example 11. Let be a -algebra in Example 5 and a of defined by Hence, is an - of as well as of .

Theorem 12. A is a of if and only if the following assertions are valid: (i) and , for all (ii), for all and (iii), for all and

Proof. Assume that Definition 2(i) is valid and , , such that and . Then, and , and so and .
Since and for all , it follows from (i) that and so that and for all . Assume that Definition 9 holds.
Let , and , be such that , , and . Then, and . It follows from Definition 9: So that and .
Again, suppose that (ii) and (iii) are valid. Also, for every and . Hence, and by (ii) and (iii), respectively, and thus,

Theorem 13. A is an -BFBI of if and only if it satisfies the subsequent assertions: (i) and (ii)(iii)for all .

Proof. Suppose be an -BFBI of . Let be such that and . If and , and for every and