Generalized Hesitant Fuzzy Ideals in Semigroups

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 (m, n)-ideals and hesitant fuzzy (m, 0)-ideals (resp. (0, n)-ideals) are introduced. Furthermore, conditions for a hesitant fuzzy (m, n)-ideal ((m, 0)-ideal, (0, n)-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 (m, n)-ideals ((m, 0)-ideals, (0, n)-ideals) are obtained. Also, the characterizations of different classes of semigroups in terms of their hesitant fuzzy (m, n)-ideals and hesitant fuzzy (m, 0)-ideals ((0, n)-ideals) are investigated.


Introduction
e 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. e 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. is is really helpful to express the hesitancy of people in everyday life. e 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. e 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][4][5][6][7][8][9]). Also, fuzzy ideal theory of algebraic structures has been studied on various aspects in [10][11][12][13].

Preliminaries
A nonempty set S endowed with an associative binary operation is called a semigroup. roughout our discussion, S will denote a semigroup unless otherwise mentioned.
A subset ∅ ≠ Ω of S is called a sub-semigroup of S if ΩΩ ⊆ Ω, and Ω is called the left (resp. right) ideal of S if SΩ ⊆ Ω (resp. ΩS ⊆ Ω). If Ω is both left and right ideals of S, Let R be a reference set. en, we define the hesitant fuzzy set (HFS) on R in terms of a function F H such that when applied to R, it returns a subset of [0, 1].
For a HFS F H on S and Z, κ ∈ S, we use the notations Two HFSs F H and F H ∩ G H are defined as follows: For Ω⊆S, we denote by χ H Ω the hesitant characteristic fuzzy set of Ω, which is defined as We denote the identity HFS by I H S , and it is defined as follows: Let A, B⊆S. en, we have roughout the paper, ℘ H R , ℘ H L , and ℘ H B will stand for the set of all hesitant fuzzy right ideals, hesitant fuzzy left ideals, and hesitant fuzzy right bi-ideals of S. e concept of (m, n)-ideals of semigroups was given by Lajos [36]. Also, the study of (m, n)-ideals in different algebraic structures has been conducted by several authors [37][38][39][40][41][42][43]. A sub-semigroup A of S is called an (m, n)-ideal of S [36] if A m SA n ⊆A, where m and n are nonnegative integers.

Main Results
roughout the paper, ℘ H(m,n) will stand for the set of all hesitant fuzzy (m, n)-ideals of S.
We illustrate it by the following example.

2
Journal of Mathematics a semigroup with the following multiplication table: Let F H 1 and F H 2 be two HFS of S such that Proof. (⇒) Let r 1 , r 2 , . . . , r m , z, s 1 , s 2 , . . . , s n ∈ S. en, the following are observed.
where T ∈ P([0, 1]), is said to be a hesitant T-level subset of F H .

Lemma 4.
Let F H be the HFS of (m, n)-regular semigroup S.
Proof. Suppose that F H ∈ ℘ H(m,n) and r, Z, s ∈ S. Since S is (m, n)-regular, rZs � r m pr n Zs m qs n for some p, q ∈ S. We have F H (rZs) � F H r m pr n Zs m qs n � F H r m pr n Zs m q s n as required. Proof. (⇒) Take any Z ∈ S. en, Z � Z m ℓZ n for some ℓ ∈ S.
We have is implies that there exist elements u, v in S with Z � uv such that So, , and it follows that κ ∈ B n and u ∈ B n . Since a � Zκ and Z � uv, therefore, a � Zκ � uvκ ∈ B m SB n . us, B⊆B m SB n . erefore, B � B m SB n . Hence, by eorem 2of [44], S is (m, n)-regular. □ Lemma 6. If F H ∈ ℘ H(m,n) and G H is a HFSS of S such that for all r 1 , r 2 , . . . , r m , Z ∈ S. Dually, a hesitant fuzzy (0, n)-ideal of S can be defined.
Lemma 9. In S, the following assertions hold: Hence, F H ∈ ℘ H R .
e following assertions are true in S: (⇐) Let a ∈ R ∩ L for R ∈ I (m,0) and L ∈ I (0,n) . By Lemma 10, χ H R ∈ ℘ H(m,0) and χ H L ∈ ℘ H(0,n) . erefore, by hypothesis, , and it follows that x ∈ R m , y ∈ L and u ∈ R, v ∈ L n . As a � xy and a � uv, a � xy ∈ R m L and a � uv ∈ RL n imply a ∈ R m L ∩ RL n . us, we obtain R ∩ L⊆R m L ∩ RL n . Also, R m L ∩ RL n ⊆R ∩ L.
as required.