Generalized Hesitant Fuzzy Ideals in Semigroups

Muhiuddin, G.; Alanazi, A. M.; Mahboob, A.; Al-Kadi, D.

doi:https://doi.org/10.1155/2020/8856287

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8856287 | https://doi.org/10.1155/2020/8856287

Generalized Hesitant Fuzzy Ideals in Semigroups

G. Muhiuddin,¹A. M. Alanazi,¹A. Mahboob,²and D. Al-Kadi³

Academic Editor: Hee S. Kim

Received06 Sept 2020

Accepted17 Nov 2020

Published22 Dec 2020

Abstract

In this paper, as a generalization of the concepts of hesitant fuzzy bi-ideals and hesitant fuzzy right (resp. left) ideals of semigroups, the concepts of hesitant fuzzy -ideals and hesitant fuzzy -ideals (resp. -ideals) are introduced. Furthermore, conditions for a hesitant fuzzy -ideal (-ideal, -ideal) to be a hesitant fuzzy bi-ideal (right ideal, left ideal) are provided. Moreover, several correspondences between bi-ideals (right ideals, left ideals) and hesitant fuzzy -ideals (-ideals, -ideals) are obtained. Also, the characterizations of different classes of semigroups in terms of their hesitant fuzzy -ideals and hesitant fuzzy -ideals (-ideals) are investigated.

1. Introduction

The fuzzy set theory introduced by Zadeh has been applied to different fields. Furthermore, in the literature, a number of generalizations and extensions of fuzzy sets have been introduced, for instance, intuitionistic fuzzy sets, interval-valued fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. As a new generalization of fuzzy sets, Torra [1] introduced the notion of hesitant fuzzy sets which permit the membership degree of an element to a set to be represented by a set of possible values between 0 and 1 (see [1, 2]). Torra [1] defined hesitant fuzzy sets in terms of a function that returns a set of membership values for each element in the domain. The hesitant fuzzy set offers a more accurate representation of hesitancy among people in expressing their preferences over objects than the fuzzy set or its classical extensions. This is really helpful to express the hesitancy of people in everyday life. The hesitant fuzzy set is a valuable tool to deal with uncertainty, which can be accurately and ideally described in terms of decision makers’ opinions.

Torra [1] defined hesitant fuzzy sets as a function returning a collection of membership values for each domain element. The hesitant fuzzy set offers a more accurate representation of hesitancy among people in expressing their preferences over objects than the fuzzy set or its classical extensions. Fuzzy set theory has been applied to different classes in semigroups (see, for e.g., [3–9]). Also, fuzzy ideal theory of algebraic structures has been studied on various aspects in [10–13].

Hesitant fuzzy set theory was applied to many practical problems, particularly in the field of decision-making (see, for e.g., [1, 2, 14–19]). Later on, Jun and Song applied the notion of hesitant fuzzy sets to -algebras and EQ-algebras (see [20, 21]). Recently, hesitant fuzzy set theory has been applied to various algebraic structures on different aspects, namely, Jun et al. applied the hesitant fuzzy set theory to -algebras and semigroups (see [22–25]), and Muhiuddin et al. applied the hesitant fuzzy set theory to residuated lattices, lattice implication algebras, and -algebras (see [26–35]). Motivated by a lot of work on hesitant fuzzy sets, we introduce the notions of hesitant fuzzy -ideals, hesitant fuzzy -ideals, and hesitant fuzzy -ideals of a semigroup by generalizing the concept of hesitant fuzzy bi-ideals, hesitant fuzzy right ideals, and hesitant fuzzy left ideals. Furthermore, associated properties of these generalized notions are discussed. Moreover, characterizations of different semigroup classes such as -regular, -regular, and -regular semigroups in terms of their hesitant fuzzy -ideals, hesitant fuzzy -ideals, and hesitant fuzzy -ideals are given.

2. Preliminaries

A nonempty set endowed with an associative binary operation is called a semigroup. Throughout our discussion, will denote a semigroup unless otherwise mentioned.

A subset of is called a sub-semigroup of if , and is called the left (resp. right) ideal of if . If is both left and right ideals of , then it is called an ideal of . A sub-semigroup of is called a bi-ideal of if .

Let be a reference set. Then, we define the hesitant fuzzy set (HFS) on in terms of a function such that when applied to , it returns a subset of .

For a HFS on and , we use the notations and .

Two HFSs and are defined as follows:respectively.

For any HFSs and on , we define if .

For any two HFSs and of , the HFS is defined as

For , we denote by the hesitant characteristic fuzzy set of , which is defined as

We denote the identity HFS by , and it is defined as follows:

Let . Then, we have (1) . (2) .

A HFS is called a hesitant fuzzy sub-semigroup (briefly HFSS) of if , , and is called a hesitant fuzzy left (resp. right) ideal (briefly HFLI and HFRI) of if , . If is both HFLI and HFRI of , then it is called a hesitant fuzzy ideal of . A HFSS is called a hesitant fuzzy bi-ideal (briefly HFBI) of if for each .

Throughout the paper, , and will stand for the set of all hesitant fuzzy right ideals, hesitant fuzzy left ideals, and hesitant fuzzy right bi-ideals of .

The concept of -ideals of semigroups was given by Lajos [36]. Also, the study of -ideals in different algebraic structures has been conducted by several authors [37–43]. A sub-semigroup of is called an -ideal of [36] if , where are nonnegative integers. Here, .

The set of all -ideals, -ideals, and -ideals will be denoted by , and.

3. Main Results

Definition 1. A HFSS of is called a hesitant fuzzy -ideal of if for all .
Throughout the paper, will stand for the set of all hesitant fuzzy -ideals of .

Lemma 1. Let . Then, .

Proof. Straightforward.

Remark 1. Let . Then, in general. We illustrate it by the following example.

Example 1. Let be a semigroup with the following multiplication table:
Let and be two HFS of such thatThen, but because .

Lemma 2. Let . Then, .

Proof. () Let . Then, the following are observed. Case 1: if for any , then Case 2: if for any , then Case 3: if and , then Case 4: if , then . Therefore, Hence, . () Let . If , then implies . Therefore, . Thus, .

Definition 2. For any HFS of , the setwhere , is said to be a hesitant -level subset of .

Theorem 1. Let be the HFS of . Then, the hesitant -level subset , provided .

Proof. () For any and suppose, to the contrary, that . Then, there exists such that . So, , but , a contradiction. Thus, for all . Hence, . () Let and , where . Then, and . By definition, . Therefore, . Hence, .

Theorem 2. Let be the HFSS of S. Then, .

Proof. () Let and . If , then . If , then there exist in such that , , and . We have the following. Case 1: when , then such that implies and such that and such that and such that and such that and Case 2: when , then such that and such that and such that and such that and such that and Now, we have () Assume that . For any , let . Since , we have Hence, .

Lemma 3. Let be the HFS of . Then, .

Proof. Straightforward.

Remark 2. In general, .

Example 2. Let be a semigroup with the following multiplication table:
Define the HFS of as follows: and . Then, , but .

Definition 3. A semigroup is called -regular if such that .

Lemma 4. Let be the HFS of -regular semigroup . Then, .

Proof. Suppose that and . Since is -regular, for some . We haveas required.

Lemma 5. Let be the HFS. Then, for any and .

Proof. Let . As , we have

Theorem 3. is -regular for each of .

Proof. () Take any . Then, for some . We have Therefore, . Suppose that . Since is the HFS of , by hypothesis, . Therefore, . Hence, is -regular.□

Theorem 4. is -regular

Proof. () Let . Then, by hypothesis and Theorems 2 and 4, and . Hence, . () Let and , so by hypothesis, we have implies . This implies that there exist elements in with such that This implies that there exist elements in with such that So, and , and it follows that . Since and , therefore, . Thus, . Therefore, . Hence, by Theorem 2of [44], is -regular.

Lemma 6. If and is a HFSS of S such thatthen .

Proof. Since is a HFSS of , by Theorem 2, it is sufficient to show that . Now,Hence, .

Lemma 7. Let and be a HFS of . If or , then(1).(2).

Proof. When , we have Therefore, is a HFSS of . Also, we have Thus, . Similarly, when , then . (2)Similar to (1).

4. Hesitant Fuzzy -Ideals and Hesitant Fuzzy -Ideals

Definition 4. A HFSS of is called a hesitant fuzzy -ideal of iffor all .
Dually, a hesitant fuzzy -ideal of can be defined.

Lemma 8. Let be the HFS of . Then, (resp. (resp. ) .

Proof. Straightforward.

Remark 3. In general, converse of Lemma 8 does not hold.

Example 3. In Example 2, the HFS , but

Definition 5. A semigroup is called -regular (resp. -regular) if such that .

Lemma 9. In , the following assertions hold:(1)In -regular semigroup , (2)In -regular semigroup ,

Proof. Let and . Since is -regular, such that . Therefore, we have Hence, . (2) Similar to the proof of (1).

Lemma 10. Let . Then, (resp. ) HFS (resp. ).

Proof. () Let . If for any , then If for each , then . Therefore, Hence, . () Suppose that . If , then implies . Therefore, . Thus, .

Theorem 5. Let be the HFS of . Then, (resp. ) , provided (resp. ).

Proof. () Suppose that and , where . Then, . By Definition 4, . Therefore, . Hence, . () Let and . Suppose, to the contrary, that . Then, there exists such that . This implies that , but , a contradiction. Thus, for all . Hence, by Definition 4, is a hesitant fuzzy -ideal of .

Theorem 6. Let be any HFSS of . Then, (resp. ) (resp. ).

Proof. On the similar lines to the proof of Theorem 2.

Lemma 11. If is an -regular semigroup, then and .

Proof. Let be an -regular semigroup and . Then, . As is -regular, we haveand so, we obtain . Hence, . Similarly, we may prove that .

Theorem 7. The following statements hold in S:(1) is -regular for each HFS of (2) is -regular for each HFS of

Proof. (1)() Take any . Then, such that . Now, we have Therefore, . () Let . Since is the HFS of , by hypothesis, . So, , and hence, is -regular.(2)Similar to the proof of (1).

Theorem 8. The following assertions are true in S:(1) is -regular (2) is -regular

Proof. (1)() Let . Then, by hypothesis and Theorems 7 and 6, we have and . Hence, . () Let be any -ideal of , and take . Then, by hypothesis, we have implies . Therefore, there exist elements in with such that and . As we have , , and it follows that . Since , therefore, . Thus, . Since is -ideal of , . Therefore, . Hence, by Theorem 1 of [44], is -regular.(2)Similar to the proof of (1).

Theorem 9. A semigroup is -regular and .

Proof. () Let and . As is -regular, we have ,and so, . Similarly, . Thus, . As and , . Therefore, . () Let for and . By Lemma 10, and . Therefore, by hypothesis, Since and , , and . This implies that there exist in with such that , and , and it follows that and . As and , and imply . Thus, we obtain . Also, . Therefore, . Hence, by Theorem 3 of [44], is -regular.

Proposition 1. Let and . If , then the product .

Proof. Let . Then, we haveTherefore, is a HFSS of . Also, we haveas required.

Lemma 12. Let be a HFS of . Then, .

Proof. Straightforward.

Lemma 13. If is -regular, then there exist and such that .

Proof. Let . Then, . As is -regular, . Therefore, . Let and . By Lemma 12, and . As is -regular, and . Thus,as required.

Lemma 14. If is -regular, then , and for each HFS of , .

Proof. Let and be the HFS of . We haveTherefore, .
By Lemmas 13 and 14, we have the following.

Corollary 1. If is -regular, then there exist and such that .

5. Conclusion

The principal objective of this paper is to establish the notions of the hesitant fuzzy - ideal, hesitant fuzzy -ideal, and hesitant fuzzy -ideal and to improve the understanding of various semigroup classes through the use of these notions. In particular, if we take in the hesitant fuzzy -ideal, hesitant fuzzy -ideal, and hesitant fuzzy -ideal, then we get the hesitant fuzzy bi-ideal, hesitant fuzzy right ideal, and hesitant fuzzy left ideal. The concepts presented in this paper are therefore more general. Furthermore, if we put in the results of this paper, then most of the results of the paper [? ] are deduced as corollaries which are the key application of the findings of this paper and a proof of the genuineness of the notions presented in this paper.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the manuscript.

Acknowledgments

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

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Copyright © 2020 G. Muhiuddin et al. 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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