A Hesitant Fuzzy Set Approach to Ideal Theory in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.4652pt" version="1.1" viewBox="-0.0657574 -11.3994 8.86425 11.4652" width="8.86425pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-132" d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"/></g></svg>-Semigroups

Abbasi, Mohammad Y.; Talee, Aakif F.; Khan, Sabahat A.; Hila, Kostaq

doi:https://doi.org/10.1155/2018/5738024

Advances in Fuzzy Systems

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Research Article | Open Access

Volume 2018 | Article ID 5738024 | https://doi.org/10.1155/2018/5738024

A Hesitant Fuzzy Set Approach to Ideal Theory in -Semigroups

Mohammad Y. Abbasi,¹Aakif F. Talee,¹Sabahat A. Khan,¹and Kostaq Hila²

Academic Editor: Jose A. Sanz

Received14 May 2018

Revised25 Jul 2018

Accepted14 Aug 2018

Published02 Sept 2018

Abstract

We apply the hesitant fuzzy sets theory to -semigroups and provide some characterizations of hesitant fuzzy left (right and bi-) ideals. We introduce the hesitant fuzzy left (resp., right and two-sided) ideal, hesitant fuzzy bi-ideal, and hesitant fuzzy interior ideal in -semigroup and study some properties of them. Finally, a characterization of a simple -semigroup by means of a hesitant fuzzy simple -semigroup is obtained.

1. Introduction

The concept of a fuzzy set, introduced by Zadeh [1], provides a natural framework for generalizing some of the notions of classical algebraic structures. After the introduction of the concept of fuzzy sets by Zadeh, several researchers conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer science, artificial intelligence, control engineering, expert, robotics, automata theory, finite state machine, graph theory logic, and many branches of pure and applied mathematics (cf. [2]). Fuzzy set theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. But in fuzzy sets theory, there is no means to incorporate the hesitation or uncertainty in the membership degrees and the fuzzy set fails in cases when it has to manage the insufficient understanding of membership degrees. For this, different extensions and generalizations of fuzzy sets have been introduced, such as (i) Atanassov’s intuitionistic fuzzy sets (IFS) [3] which give both a membership degree and a nonmembership degree, (ii) type-2 fuzzy sets (T2FS) [4] which let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems and that incorporate uncertainty about the membership function into fuzzy set theory (the value of membership function is given by a fuzzy set), (iii) interval-valued fuzzy sets (IVFS) [4, 5] in which the values of the membership degrees are intervals of numbers instead of the numbers, and thus, i-v fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets, and (iv) fuzzy multisets [6] based on multisets, that is, fuzzy subsets whose elements may occur more than one time (an element of a fuzzy multiset can occur more than once with possibly the same or different membership values). As a new generalization of fuzzy sets, the so-called hesitant fuzzy sets (briefly HFSs) has been introduced by Torra [7–9] as more suitable tools for dealing with group decision-making problems when experts have a hesitation among several possible memberships for an element to a set, rather than a margin of error considered in intuitionistic fuzzy sets (IFSs) or some possibility distribution on the possible values considered in type-2 fuzzy sets (cf. [10]). The HFS maps the membership degree of an element to a set presented as several possible values between zero and one, which can better describe the situations where people have hesitancy in providing their preferences over objects in the process of decision-making and to get the optimal alternative in a decision-making problem with multiple attributes and multiple persons. During the evaluating process in practice, however, these possible memberships may be not only crisp values in , but also interval values. HFS theory has been applied to different algebraic structures. The structure of semigroups containing hesitant fuzzy ideals was studied by Jun et al. [11–13]. Abbasi et al. [14, 15] applied the notion of HFSs to po-semigroups. Ali et al. [16] studied the notion of HFSs in AG-Groupoids.

In 1981, Sen [17] introduced the concept and notion of the -semigroup as a generalization of plain semigroup and ternary semigroup. Many classical notions and results of the theory of semigroups have been extended and generalized to -semigroups, by many mathematicians, for instance, Dutta and Adhikari [18, 19], Saha and Sen [20–22], and Hila [23–25]. Mustafa et al. [26] have introduced the - semigroup in terms of intuitionistic fuzzy sets. Sardar, Majumder, Dutta, Davvaz, and Hila [27–30] studied the -semigroup in terms of fuzzy subsets. Considering all the reasons described above about the problems faced with several generalizations of fuzzy sets in different algebraic structures, which led to the introduction of HFS as a better tool, in this paper, our main aim is to apply the HFS theory to -semigroups as a more general algebraic structure. We introduce the hesitant fuzzy left (resp., right and two-sided) ideal, hesitant fuzzy bi-ideal, and hesitant fuzzy interior ideal in -semigroup and study some properties of them. Finally, a characterization of a simple -semigroup by means of a hesitant fuzzy simple -semigroup is obtained.

2. Preliminaries

In this section, we introduce some necessary notions on -semigroups and present some operations on HFSs that will be used throughout the paper.

We first recall the definition of the -semigroup as a generalization of semigroup and ternary semigroup in another way as follows (cf. [17, 22, 24]).

Definition 1. Let and be two nonempty sets. Denote by the letters of the English alphabet the elements of and with the letters of the Greek alphabet the elements of . Any map from will be called a -multiplication in and denoted by . The result of this multiplication for and is denoted by . A -semigroup is an ordered pair where and are nonempty sets and is a -multiplication on which satisfies the following property: .

This kind of structure is often called one sided -semigroup or simply -semigroup.

Example 2. Let be a semigroup and be any nonempty set. Define a mapping by for all and . Then is a -semigroup.

Example 3. Let be the set of all matrices with entries from a field, where are positive integers. Let be the power set of . Then it is easy to see that is not a semigroup under multiplication of matrices because, for , the product is not defined. Let be the set of matrices with entries from the same field. Then for and , we have and since the matrix multiplication is associative, we get that is a -semigroup.

Example 4. Let be a set of all negative rational numbers. Obviously is not a semigroup under usual product of rational numbers. Let . Let and . Now if is equal to the usual product of rational numbers , then and . Hence is a -semigroup.

Example 5. Let and . Then is a -semigroup under the multiplication over complex numbers while is not a semigroup under complex number multiplication.

These examples show that every semigroup is a -semigroup. Therefore, -semigroups are a generalization of semigroups.

A nonempty subset of a -semigroup is called a sub--semigroup of if for all and , .

Example 6. Let and . Then is a -semigroup under usual multiplication. Let . We have that is a nonempty subset of and for all and . Then is a sub--semigroup of .

Definition 7. The element of a -semigroup is called regular in if , where . is called regular if and only if every element of is regular.

The following is the definition of the so-called both-sided -semigroup [18].

Definition 8. Let and be two nonempty sets. Denote by the letters of the English alphabet the elements of and with the letters of the Greek alphabet the elements of . Any map from will be called a -multiplication in and denoted by . The result of this multiplication for and is denoted by . Any map from will be called a -multiplication in and denoted by . The result of this multiplication for and is denoted by . A both-sided -semigroup is an ordered triple where and satisfy the following property: ,

It is clear that a both-sided -semigroup is always a one sided -semigroup, while the converse does not hold in general.

Clearly, a both-sided -semigroup is regular if and only if , for all [31].

Throughout this paper unless or otherwise mentioned stands for one sided -semigroup and call it simply -semigroup.

Let be a reference set. A HFS on is a function that when applied to returns a finite subset of values in : where denotes the set of all subsets of .

Let and be any two HFSs on . We define if , for all .

Let be a -semigroup. For any HFSs and in , the hesitant fuzzy product of and is defined to be the HFS on as follows:

For any two HFSs and on , the hesitant union of and is defined to be HFS on as follows: and the hesitant intersection of and is defined to be HFS on as follows:

Let be any nonempty subset of . Recall that, we denote by the characteristic HFS on as follows:

Let be a -semigroup. We define a hesitant fuzzy subset of as follows:

3. Main Results

Definition 9. Let be a -semigroup and be a HFS on . Then is called a hesitant fuzzy -semigroup on if it satisfies ;a hesitant fuzzy left ideal on if it satisfies ;a hesitant fuzzy right ideal on if it satisfies ;a hesitant fuzzy ideal on if is both a hesitant fuzzy left ideal and a hesitant fuzzy right ideal on ;a hesitant fuzzy bi-ideal on if it satisfies .

Example 10. Let = be a set of all matrices over the field and = be a set of all matrices over the field . Then is a -semigroup with respect to usual matrix products and , for all and .
Define a hesitant fuzzy subset of such that Clearly, is a hesitant fuzzy ideal of .

Example 11. Let be a set of nonpositive integers and be a set of nonpositive even integers. Then is a -semigroup with respect to usual multiplication.
Define a hesitant fuzzy subset of such that Clearly, is a hesitant fuzzy ideal of .

Example 12. Let = and be two nonempty sets. Clearly is a -semigroup with respect to the operation defined below:

Let be a hesitant fuzzy subset of such that

Clearly is a hesitant fuzzy bi-ideal of . But is neither a hesitant fuzzy right ideal nor a hesitant fuzzy left ideal of , since and , respectively.

Theorem 13. Let be a nonempty hesitant fuzzy subset of a -semigroup Then the following conditions are equivalent:
is a hesitant fuzzy bi-ideal of .
and

Proof. Let be a hesitant fuzzy bi-ideal of . Then is a hesitant fuzzy subsemigroup of . Let . If some and such that , then we have Thus . Otherwise, (. Hence
Let we prove . For this, let and let us suppose there exist and such that and Since is a hesitant fuzzy bi-ideal of , then we have Thus,
Otherwise, (. Hence .
(1). Assume that . Let and such that . Then we have Thus, is a hesitant fuzzy -subsemigroup of . Now let and such that . Since , then it follows that for some . Hence, is a hesitant fuzzy bi-ideal of .

Theorem 14. Let be a nonempty hesitant fuzzy -subsemigroup of a -semigroup Then the following conditions are equivalent:
is a hesitant fuzzy left (resp., right) ideal of .
(resp., ).

Proof. Let be a hesitant fuzzy left ideal of and . Suppose there exist and such that . Then we have Since is a hesitant fuzzy left ideal of , then it follows that, for all and , So in particular, , for all Thus .
If there do not exist and such that , then . Hence .
. Assume that . Let , , and . Then we have Hence, is a hesitant fuzzy left ideal of . Similarly, we can prove the other case.

Theorem 15. Let be a regular both-sided -semigroup, be a hesitant fuzzy right ideal, and be a hesitant fuzzy left ideal . Then .

Proof. Let be a hesitant fuzzy right ideal and be a hesitant fuzzy left ideal of . Let Suppose there exist and such that . Then we have If there do not exist such that , then
Thus .
For the reverse inclusion, let Since is regular, then there exist and such that , where Therefore, Thus . Hence .

Let be a nonempty hesitant fuzzy subset of a -semigroup and . Then the set is called the -cut of .

Remark 16. Let be a hesitant fuzzy ideal of a -semigroup and such that . Then .

Theorem 17. Let be a nonempty hesitant fuzzy subset of a -semigroup . Then the -cut of is a left (right) ideal of for every , provided it is nonempty if and only if is a hesitant fuzzy left (right) ideal of

Proof. For every , let be a left ideal of . We first show that is a hesitant fuzzy -subsemigroup of . If possible there exist such that . Let Then . Thus but , a contradiction. Thus, . Hence is an hesitant fuzzy -subsemigroup of . Again suppose that there exist such that . Since , then let . Thus, , but , a contradiction. Thus, . Hence is a hesitant fuzzy left ideal of
Conversely, suppose that is a hesitant fuzzy left ideal of and such that is nonempty. Let and . Then and which implies . Since is a hesitant fuzzy ideal, then it is a hesitant fuzzy -subsemigroup, and hence . Consequently, . Hence is a -subsemigroup of . Now, let and . Then and so . Hence is a left ideal of . Similarly, we can prove the other case.

Definition 18. A hesitant fuzzy -subsemigroup on -semigroup is called a hesitant fuzzy interior ideal on if it satisfies

Theorem 19. Let be a nonempty hesitant fuzzy subset of a -semigroup . Then the -cut of is an interior ideal of for every , provided it is nonempty if and only if is a hesitant fuzzy interior ideal of

Proof. For every , let be an interior ideal of . Assume that there exist such that . Since , then let . Thus, , but , a contradiction. Hence, .
Conversely, assume that is a hesitant fuzzy interior ideal of and such that is nonempty. Let and . Then and so .
The rest of the proof is a consequence of Theorem 17.

It is well-known that, in a -semigroup , every hesitant fuzzy two-sided ideal is a hesitant fuzzy interior ideal of , but the converse is not true in general. The following example shows that the converse of this property does not hold in general.

Example 20. Let be a set of nonpositive integers and be a set of nonpositive even integers. Then is a -semigroup with respect to usual multiplication.
Define a hesitant fuzzy subset of such that

Clearly, is a hesitant fuzzy interior ideal of . But is not hesitant fuzzy ideal of , since .

Now we will show that, in a regular -semigroup, hesitant fuzzy ideals and the hesitant fuzzy interior ideals coincide.

Theorem 21. Let be a HFS in a regular -semigroup . Then is a hesitant fuzzy ideal of if and only if is a hesitant fuzzy interior ideal of .

Proof. Let a hesitant fuzzy ideal of . For any , , , and , we have Hence is a hesitant fuzzy interior ideal of .
Conversely, let and . Since is regular, then there exist elements such that = . Since is a hesitant fuzzy interior ideal of , then we have . So is a hesitant fuzzy right ideal of Similarly, we can prove that is a hesitant fuzzy left ideal of Hence is a hesitant fuzzy ideal of

In order to conclude the paper, we obtain the following characterization of a simple -semigroup by means of a hesitant fuzzy simple -semigroup.

Definition 22. A -semigroup is said to be simple if it does not contain any proper ideal. A -semigroup is said to be hesitant fuzzy simple if every hesitant fuzzy ideal of is a constant function.

Theorem 23. A -semigroup is simple if and only if it is hesitant fuzzy simple.

Proof. Suppose that the -semigroup is simple. Let be a hesitant fuzzy ideal of and . Then, by Theorem 17, and are ideals of . Since is simple, then . Therefore, and . In particular, and , hence and . Thus, , for all . Hence is a constant function. Consequently, is a hesitant fuzzy simple.
Conversely, assume that is a hesitant fuzzy simple. Let be any ideal of . Then its characteristic function is a hesitant fuzzy ideal of and thus is a constant function. Let . Since is nonempty, then and so . Thus we obtain . Hence is simple.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2018 Mohammad Y. Abbasi 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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