# Generalized Hesitant Fuzzy Ideals in Semigroups

Authors: G. Muhiuddin; A. M. Alanazi; A. Mahboob; D. Al-Kadi
Journal: Journal of Mathematics
 (2020)

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## 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 fuzzym,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.


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## Body

## 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 MTL-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 BCK/BCI-algebras and semigroups (see [22–25]), and Muhiuddin et al. applied the hesitant fuzzy set theory to residuated lattices, lattice implication algebras, and BCK/BCI-algebras (see [26–35]). Motivated by a lot of work on hesitant fuzzy sets, we introduce the notions of hesitant fuzzy m,n-ideals, hesitant fuzzy m,0-ideals, and hesitant fuzzy 0,n-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 m,n-regular, m,0-regular, and 0,n-regular semigroups in terms of their hesitant fuzzy m,n-ideals, hesitant fuzzy m,0-ideals, and hesitant fuzzy 0,n-ideals are given.


## 2. Preliminaries

A nonempty setS endowed with an associative binary operation is called a semigroup. Throughout 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⊆Ωresp.ΩS⊆Ω. If Ω is both left and right ideals of S, then it is called an ideal of S. A sub-semigroup Ω of S is called a bi-ideal of S if ΩSΩ⊆Ω.LetR be a reference set. Then, we define the hesitant fuzzy set (HFS) on R in terms of a function ℱℋ^ such that when applied to R, it returns a subset of 0,1.For a HFSℱℋ^ on S and ℏ,κ∈S, we use the notations ℱℏℋ^:=ℱℋ^ℏ and ℱℏℋℋ^:=ℱℋ^ℏ∩ℱℋ^κ.Two HFSsℱℋ^ and ℱℋ^∩Gℋ^ are defined as follows:(1)ℱℋ^∪Gℋ^:S⟶P0,1,ℏ↦ℱℏℋ^∪Gℏℋ^,(2)ℱℋ^∩Gℋ^:S⟶P0,1,ℏ↦ℱℏℋ^∩Gℏℋ^,respectively.For any HFSsℱℋ^ and Gℋ^ on S, we define ℱℋ^⊆Gℋ^ if ℱℏℋ^⊆Gℏℋ^∀ℏ∈S.For any two HFSsℱℋ^ and Gℋ^ of S, the HFS ℱℋ^°Gℋ^ is defined as(3)ℱℋ^°Gℋ^ℏ=∪ℏ=κℓℱℋ^κ∩Gℋ^ℓ,if there existκ,ℓ∈Ssuch thatℏ=κℓ,∅,otherwise.ForΩ⊆S, we denote by χΩℋ^ the hesitant characteristic fuzzy set of Ω, which is defined as(4)χΩℋ^ℏ=0,1ifℏ∈Ω,∅ifℏ∉Ω.We denote the identity HFS byℐSℋ^, and it is defined as follows:(5)ℐSℋ^ℏ=0,1,∀ℏ∈S.LetA,B⊆S. Then, we have(1)χAℋ^°χBℋ^=χABℋ^.(2)χAℋ^∩χBℋ^=χA∩Bℋ^.A HFSℱℋ^ is called a hesitant fuzzy sub-semigroup (briefly HFSS) of S if ∀ℏ,κ∈S, ℱℋ^ℏκ⊇ℱℋ^ℏ∩ℱℋ^κ, and ℱℋ^ is called a hesitant fuzzy left (resp. right) ideal (briefly HFLI and HFRI) of S if ∀ℏ,κ∈S, ℱℋ^ℏκ⊇ℱℋ^κresp.ℱℋ^ℏκ⊇ℱℋ^ℏ. If ℱℋ^ is both HFLI and HFRI of S, then it is called a hesitant fuzzy ideal of S. A HFSS ℱℋ^ is called a hesitant fuzzy bi-ideal (briefly HFBI) of S if ℱℋ^ℏℓκ⊇ℱℋ^ℏ∩ℱℋ^κ for each ℏ,κ,ℓ∈S.Throughout the paper,℘ℋR^,℘ℋL^, and ℘ℋB^ will stand for the set of all hesitant fuzzy right ideals, hesitant fuzzy left ideals, and hesitant fuzzy right bi-ideals of S.The concept ofm,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 [37–43]. A sub-semigroup A of S is called an m,n-ideal of S [36] if AmSAn⊆A, where mandn are nonnegative integers. Here, A0S=SA0=S.The set of allm,n-ideals, m,0-ideals, and 0,n-ideals will be denoted by ℐm,n,ℐm,0, and. ℐ0,n


## 3. Main Results

Definition 1.
A HFSSℱℋ^ of S is called a hesitant fuzzy m,n-ideal of S if ℱℋ^r1r2…rmzs1s2…sn⊇ℱr1ℋ^∩ℱr2ℋ^∩⋯∩ℱrmℋ^∩ℱs1ℋ^∩ℱs2ℋ^∩⋯∩ℱsnℋ^ for all r1,r2,…,rm,z,s1,s2,…sn∈S.
Throughout the paper,℘ℋm,n^ will stand for the set of all hesitant fuzzy m,n-ideals of S.Lemma 1.
Letℱiℋ^∈℘ℋm,n^|i∈I. Then, ∩i∈Iℱiℋ^∈℘ℋm,n^.Proof.
Straightforward.Remark 1.
Letℱiℋ^∈℘ℋm,n^|i∈I. Then, ∪i∈Iℱiℋ^∉℘ℋm,n^ in general. We illustrate it by the following example.Example 1.
LetS=ϑ,ι,κ,ℏ be a semigroup with the following multiplication table:
Letℱ1ℋ^ and ℱ2ℋ^ be two HFS of S such that(6)ℱ1ℋ^ϑ=0,0.2,ℱ1ℋ^ι=0,0.2,ℱ1ℋ^κ=∅,ℱ1ℋ^ℏ=∅,ℱ2ℋ^ϑ=0,0.2,ℱ2ℋ^ι=∅,ℱ2ℋ^κ=0,0.2,ℱ2ℋ^ℏ=∅.
Then,ℱ1ℋ^,ℱ2ℋ^∈℘ℋm,n^ but ℱ1ℋ^∪ℱ2ℋ^∉℘ℋm,n^ because ∅=ℱ1ℋ^u∪ℱ1ℋ^u=ℱ1ℋ^∪ℱ2ℋ^u=ℱ1ℋ^∪ℱ2ℋ^sc⊂ℱ1ℋ^∪ℱ2ℋ^s∩ℱ1ℋ^∪ℱ2ℋ^w=0,0.2.·ϑικℏϑϑϑϑϑιϑϑℏϑκϑϑϑϑℏϑϑϑϑLemma 2.
Let∅≠A⊆S. Then, A∈ℐm,n⇔χAℋ^∈℘ℋm,n^.Proof.
(⇒) Let r1,r2,…,rm,z,s1,s2,…,sn∈S. Then, the following are observed.Case 1: ifrı∉A for any ı∈1,2,…,m, then(7)χAℋ^r1r2…rmzs1s2…sn⊇χAℋ^r1∩χAℋ^r2∩⋯∩χAℋ^rm∩χAℋ^s1∩χAℋ^s2∩⋯∩χAℋ^sn.Case 2: ifsJ∉A for any J∈1,2,…,n, then(8)χAℋ^r1r2…rmzs1s2…sn⊇χAℋ^r1∩χAℋ^r2∩⋯∩χAℋ^rm∩χAℋ^s1∩χAℋ^s2∩⋯∩χAℋ^sn.Case 3: ifrı∉A and sJ∉A∀ı∈1,2,…,m,J∈1,2,…,n, then(9)χAℋ^r1r2…rmzs1s2…sn⊇χAℋ^r1∩χAℋ^r2∩⋯∩χAℋ^rm∩χAℋ^s1∩χAℋ^s2∩⋯∩χAℋ^sn.Case 4: ifrı,sJ∈A∀ı∈1,2,…,m,J∈1,2,…,n, then r1r2…rmzs1s2…sn∈AmSAn⊆A. Therefore,(10)χAℋ^r1r2…rmzs1s2…sn=0,1⊇χAℋ^r1∩χAℋ^r2∩⋯∩χAℋ^rm∩χAℋ^s1∩χAℋ^s2∩⋯∩χAℋ^sn.Hence,χAℋ^∈℘ℋm,n^.(⇐) Let x,z,y∈S. If x,y∈A, then χAℋ^xmzyn⊇χAℋ^x∩χAℋ^y=0,1 implies χAℋ^xmzyn=0,1. Therefore, xmzyn∈A. Thus, AmSAn⊆A.Definition 2.
For any HFSℱℋ^ of S, the set(11)ℱTℋ^=x∈S|ℱxℋ^⊇T,where T∈P0,1, is said to be a hesitant T-level subset of ℱℋ^.Theorem 1.
Letℱℋ^ be the HFS of S. Then, the hesitant T-level subset ℱTℋ^∈ℐm,n∀T∈P0,1, provided ℱTℋ^≠∅⇔ℱℋ^∈℘ℋm,n^.Proof.
(⇒) For any r1,r2,…,rm,s1,s2,…,sn∈S and suppose, to the contrary, that ℱℋ^r1∩ℱℋ^r2∩⋯∩ℱrmℋ^∩ℱs1ℋ^∩ℱs2ℋ^∩⋯∩ℱsnℋ^⊃ℱℋ^r1r2⋯rmzs1s2⋯sn. Then, there exists C∈P0,1 such that ℱr1ℋ^∩ℱr2ℋ^∩⋯∩ℱrmℋ^∩ℱs1ℋ^∩ℱs2ℋ^∩⋯∩ℱsnℋ^⊇C⊃ℱℋ^r1r2⋯rmzs1s2⋯sn. So, r1,r2,…,rm,z,s1,s2,…,sn∈ℱCℋ^, but r1r2⋯rmzs1s2⋯sn∉ℱCℋ^, a contradiction. Thus, ℱℋ^r1r2⋯rmzs1s2⋯sn⊇ℱr1ℋ^∩ℱr2ℋ^∩⋯∩ℱrmℋ^∩ℱs1ℋ^∩ℱs2ℋ^∩⋯∩ℱsnℋ^ for all r1,r2,…,rm,z,s1,s2,…,sn∈S. Hence, ℱℋ^∈℘ℋm,n^.(⇐) Let z∈S and r1,r2,…,rm,s1,s2,…,sn∈ℱTℋ^, where T∈P0,1. Then, ℱr1ℋ^⊇T,ℱr2ℋ^⊇T,…,ℱxnℋ^⊇T and ℱs1ℋ^⊇T,ℱs2ℋ^⊇T,⋯ℱymℋ^⊇T. By definition, ℱℋ^r1r2⋯rmzs1s2⋯sn⊇ℱr1ℋ^∩ℱr2ℋ^∩⋯∩ℱrmℋ^∩ℱs1ℋ^∩ℱs2ℋ^∩⋯∩ℱsnℋ^=T. Therefore, r1r2⋯rmzs1s2⋯sn∈ℱTℋ^. Hence, ℱTℋ^∈ℐm,n.Theorem 2.
Letℱℋ^ be the HFSS of S. Then, ℱℋ^∈℘ℋm,n^⇔ℱℋ^m°ℐSℋ^°ℱℋ^n⊆ℱℋ^.Proof.
(⇒) Let ℱℋ^∈℘ℋm,n^ and ℏ∈S. If ℱℋ^m°ℐSℋ^°ℱℋ^nℏ=∅, then ℱℋ^m°ℐSℋ^°ℱℋ^n⊆ℱℋ^. If ℱℋ^m°ℐSℋ^°ℱℋ^nℏ≠∅, then there exist ℓ,κ in S such that ℏ=ℓκ, ℱℋ^m°ℐSℋ^ℓ≠∅, and ℱℋ^nκ≠∅. We have the following.Case 1: whenℱℋ^m°ℐSℋ^ℓ≠∅, then∃u1,v1∈S such that ℓ=u1v1 implies ℱℋ^mu1≠∅ and ℐSℋ^v1=0,1∃u2,v2∈S such that u1=u2v2ℱℋ^u2≠∅ and ℱℋ^m−1v2≠∅∃u3,v3∈S such that v2=u3v3ℱℋ^u3≠∅ and ℱℋ^m−2v3≠∅⋮∃um−1,vm−1∈S such that vm−2=um−1vm−1ℱℋ^um−1≠∅ and ℱℋ^2vm−1=ℱℋ^m−m−2vm−1≠∅∃um,vm∈S such that vm−1=umvmℱℋ^um≠∅ and ℱℋ^vm≠∅Case 2: whenℱℋ^nκ≠∅, then∃u1′,v1′∈S such that κ=u1′v1′ℱℋ^u1′≠∅ and ℱℋ^n−1v1′≠∅∃u2′,v2′∈S such that v1′=u2′v2′ℱℋ^u2′≠∅ and ℱℋ^n−2v2′≠∅∃u3′,v3′∈S such that v2′=u3′v3′ℱℋ^u3′≠∅ and ℱℋ^n−3v3′≠∅⋮∃un−2′,vn−2′∈S such that vn−3′=un−2′vn−2′ℱℋ^un−2′≠∅ and ℱℋ^n−n−2vn−2′≠∅∃un−1′,vn−1′∈S such that vn−2′=un−1′vn−1′ℱℋ^un−1′≠∅ and ℱℋ^vn−1′≠∅Now, we have(12)ℱℋ^m°ℐSℋ^°ℱℋ^na=∪a=ℓκℱℋ^m°ℐSℋ^ℓ∩ℱℋ^nκ=∪a=ℓκ∪ℓ=u1v1ℱℋ^mu1∩ℐSℋ^v1∩∪κ=u1′v1′ℱℋ^u1′∩ℱℋ^n−1v1′=∪a=ℓκ∪ℓ=u1v1∪κ=u1′v1′ℱℋ^mu1∩ℐSℋ^v1∩ℱℋ^u1′∩ℱℋ^n−1v1′=∪a=ℓκ∪ℓ=u1v1∪κ=u1′v1′ℱℋ^mu1∩ℱℋ^u1′∩ℱℋ^n−1∩v1′=∪a=ℓκ∪ℓ=u1v1∪κ=u1′v1′∪u1=u2v2ℱℋ^u2∩ℱℋ^m−1v2∩∪u1′=u2′v2′ℱℋ^u2′∩ℱℋ^n−2v2′∩ℱℋ^v1′=∪a=ℓκ∪ℓ=u1v1∪κ=u1′v1′∪u1=u2v2∪u1′=u2′v2′ℱℋ^u2∩ℱℋ^m−1v2∩ℱℋ^u2′∩ℱℋ^n−2v2′∩ℱℋ^v1′=∪a=ℓκ∪ℓ=u1v1∪κ=u1′v1′∪u1=u2v2∪u1′=u2′v2′⋯∪um−1=um,vm∪un−2′=un−1′vn−1′ℱℋ^u2∩ℱℋ^u3∩⋯∩ℱℋ^um∩ℱℋ^vm∩ℱℋ^un−1′∩ℱℋ^vn−1′∩⋯∩ℱℋ^v2′∩ℱℋ^v1′⊆∪a=ℓκℱℋ^u2u3…vmvmv1u1′u2′…un−1′vn−1′=∪a=ℓκℱℋ^ℓκsinceℓ=u2u3…vmvmv1andκ=u1′u2′…un−1′vn−1′=ℱℋ^a.(⇐) Assume that ℱℋ^m°ℐSℋ^°ℱℋ^n⊆ℱℋ^. For any r1,r2,…,rm,z,s1,s2,…,sn∈S, let a=r1r2⋯rmzs1s2⋯sn. Since ℱℋ^m°ℐSℋ^°ℱℋ^n⊆ℱℋ^, we have(13)ℱℋ^r1r2…rmzs1s2…sn=ℱℋ^a⊇ℱℋ^m°ℐSℋ^°ℱℋ^na=∪a=pqℱℋ^m°ℐSℋ^p∩ℱℋ^nq⊇ℱℋ^m°ℐSℋ^r1r2…rmz∩ℱℋ^ns1s2…sn⊇∪r1r2…rmz=uvℱℋ^mu∩ℐSℋ^u∩∪s1s2…sn=u′,v′ℱℋ^u′∩ℱℋ^n−1v′⊇ℱℋ^mr1r2…rm∩ℐSℋ^z∩ℱℋ^s1∩ℱℋ^n−1s2…yn−1sn⊇ℱℋ^mr1r2…rm∩ℱℋ^n−1s1s2…yn−1∩ℱℋ^sn⋮⊇ℱℋ^r1∩ℱℋ^r2∩⋯∩ℱℋ^rm∩ℱℋ^s1∩ℱℋ^s2∩⋯∩ℱℋ^sn.Hence,ℱℋ^∈℘ℋm,n^.Lemma 3.
Letℱℋ^ be the HFS of S. Then, ℱℋ^∈℘ℋB^⇒ℱℋ^∈℘ℋm,n^∀m,n∈ℕ.Proof.
Straightforward.Remark 2.
In general,ℱℋ^∈℘ℋm,n^⇏ℱℋ^∈℘ℋB^.Example 2.
LetS=ϑ,ι,κ,ℏ be a semigroup with the following multiplication table:
Define the HFSℱℋ^ of S as follows: ℱℋ^ϑ=ℱℋ^ℏ=0,1 and ℱℋ^ι=ℱℋ^ℏ=0. Then, ℱℋ^∈℘ℋm,n^, but ℱℋ^∉℘ℋB^.·ϑικℏϑϑϑϑϑιιιιικκκκκℏϑϑιϑDefinition 3.
A semigroupS is called m,n-regular if ∀ℏ∈S∃ℓ∈S such that ℏ=ℏmℓℏn.Lemma 4.
Letℱℋ^ be the HFS of m,n-regular semigroup S. Then, ℱℋ^∈℘ℋm,n^⇒ℱℋ^∈℘ℋB^.Proof.
Suppose thatℱℋ^∈℘ℋm,n^ and r,ℏ,s∈S. Since S is m,n-regular, rℏs=rmprnℏsmqsn for some p,q∈S. We have(14)ℱℋ^rℏs=ℱℋ^rmprnℏsmqsn=ℱℋ^rmprnℏsmqsn⊇ℱℋ^r∩ℱℋ^s,as required.Lemma 5.
Letℱℋ^ be the HFS. Then, ℱℋ^ℏ⊆ℱℋ^kℏk for any k∈ℕ and ℏ∈S.Proof.
Letℏ∈S. As ℏk=ℏℏk−1, we have(15)ℱℋ^kℏk=∪ℏk=uvfu∩ℱℋ^k−1v⊇ℱℋ^ℏ∩ℱℋ^k−1ℏk−1=ℱℋ^ℏ∩∪ℏk−1=u′v′ℱℋ^u′∩ℱℋ^k−2v′⊇ℱℋ^ℏ∩ℱℋ^ℏ∩ℱℋ^k−2ℏk−2⋮⊇ℱℋ^ℏ∩⋯∩ℱℋ^ℏ∩ℱℋ^ℏ=ℱℋ^ℏ.Theorem 3.
S is m,n-regular ⇔ℱℋ^⊆ℱℋ^m°ℐSℋ^°ℱℋ^n for each HFSℱℋ^ of S.Proof.
(⇒) Take any ℏ∈S. Then, ℏ=ℏmℓℏn for some ℓ∈S. We have(16)ℱℋ^m°ℐSℋ^°ℱℋ^nℏ=∪ℏ=rsℱℋ^m°ℐSℋ^r∩ℱℋ^ns⊇ℱℋ^m°ℐSℋ^ℏmx∩ℱℋ^nℏn=∪ℏmx=pqℱℋ^mp∩ℐSℋ^q∩ℱℋ^nℏn⊇ℱℋ^mℏm∩ℐSℋ^x∩ℱℋ^nℏn=ℱℋ^mℏm∩ℱℋ^nℏn⊇ℱℋ^ℏ∩ℱℋ^ℏby Lemma2.14=ℱℋ^ℏ.Therefore,ℱℋ^⊆ℱℋ^m°ℐSℋ^°ℱℋ^n.⇐ Suppose that ℏ∈S. Since χℏℋ^ is the HFS of S, by hypothesis, χℏℋ^⊆χℏℋ^m°ℐSℋ^°χℏℋ^n=χℏmSℏnℋ^. Therefore, ℏ∈ℏmSℏn. Hence, S is m,n-regular.□Theorem 4.
S is m,n-regular ⇔ℱℋ^=ℱℋ^m°ℐSℋ^°ℱℋ^n∀ℱℋ^∈℘ℋm,n^Proof.
(⇒) Let ℱℋ^∈℘ℋm,n^. Then, by hypothesis and Theorems 2 and 4, ℱℋ^m°ℐSℋ^°ℱℋ^n⊆ℱℋ^ and ℱℋ^⊆ℱℋ^m°ℐSℋ^°ℱℋ^n. Hence, ℱℋ^=ℱℋ^m°ℐSℋ^°ℱℋ^n.(⇐) Let B∈ℐm,n and b∈B, so by hypothesis, we have χBℋ^m°ℐSℋ^°χBℋ^nb=χAb=1 implies χBℋ^m°ℐSℋ^°χBℋ^nb=1. This implies that there exist elements ℏ,κ in S with b=ℏκ such that(17)χBℋ^m°ℐSℋ^ℏ=0,1andχBℋ^nκ=0,1.This implies that there exist elementsu,v in S with ℏ=uv such that(18)χBℋ^mu=0,1andℐSℋ^v=0,1.So,0,1=χBℋ^nκ=χBnℋ^κ and 0,1=χBℋ^mu=χBmℋ^u, and it follows that κ∈Bnandu∈Bn. Since a=ℏκ and ℏ=uv, therefore, a=ℏκ=uvκ∈BmSBn. Thus, B⊆BmSBn. Therefore, B=BmSBn. Hence, by Theorem 2of [44], S is m,n-regular.Lemma 6.
Ifℱℋ^∈℘ℋm,n^ and Gℋ^ is a HFSS of S such that(19)ℱℋ^m°ℐSℋ^°ℱℋ^n⊆Gℋ^⊆ℱℋ^,then Gℋ^∈℘ℋm,n^.Proof.
SinceGℋ^ is a HFSS of S, by Theorem 2, it is sufficient to show that Gℋ^m°ℐSℋ^°Gℋ^n⊆Gℋ^. Now,(20)Gℋ^m°ℐSℋ^°Gℋ^na⊆ℱℋ^m°ℐSℋ^°ℱℋ^na⊆Gℋ^a.
Hence,Gℋ^∈℘ℋm,n^.Lemma 7.
Letℱℋ^∈℘ℋm,n^ and Gℋ^ be a HFS of S. If ℱℋ^°Gℋ^⊆ℱℋ^ or Gℋ^°ℱℋ^⊆ℱℋ^, then(1)
ℱℋ^°Gℋ^∈℘ℋm,n^.(2)
Gℋ^°ℱℋ^∈℘ℋm,n^.Proof.
Whenℱℋ^°Gℋ^⊆ℱℋ^, we have(21)ℱℋ^°Gℋ^°ℱℋ^°Gℋ^=ℱℋ^°ℱℋ^°Gℋ^=ℱℋ^°ℱℋ^°Gℋ^⊆ℱℋ^°Gℋ^.Therefore,ℱℋ^°Gℋ^ is a HFSS of S. Also, we have(22)ℱℋ^°Gℋ^m°S°ℱℋ^°Gℋ^n=ℱℋ^°Gℋ^m°S°ℱℋ^°Gℋ^n−1°ℱℋ^°Gℋ^⊆ℱℋ^m°S°ℱℋ^n−1°ℱℋ^°Gℋ^⊆ℱℋ^m°Gℋ^°ℱℋ^n°Gℋ^⊆ℱℋ^°Gℋ^.Thus,ℱℋ^°Gℋ^∈℘ℋm,n^. Similarly, when Gℋ^°ℱℋ^⊆ℱℋ^, then ℱℋ^°Gℋ^∈℘ℋm,n^.(2)Similar to (1).


## 4. Hesitant Fuzzym,0-Ideals and Hesitant Fuzzy 0,n-Ideals

Definition 4.
A HFSSℱℋ^ of S is called a hesitant fuzzy m,0-ideal of S if(23)ℱℋ^r1r2…rmℏ⊇ℱr1ℋ^∩ℱr2ℋ^∩⋯∩ℱrmℋ^,for all r1,r2,…,rm,ℏ∈S.
Dually, a hesitant fuzzy0,n-ideal of S can be defined.Lemma 8.
Letℱℋ^ be the HFS of S. Then, ℱℋ^∈℘ℋR^ (resp. ℱℋ^∈℘ℋL^⇒ℱℋ^∈℘ℋm,0^ (resp. ℱℋ^∈℘ℋ0,n^) ∀m,n∈ℕ.Proof.
Straightforward.Remark 3.
In general, converse of Lemma8 does not hold.Example 3.
In Example2, the HFS ℱℋ^∈℘ℋm,0^,℘ℋ0,n^∀m,n∈ℕ,m,n≥2, but ℱℋ^∉℘ℋR^,℘ℋL^Definition 5.
A semigroupS is called m,0-regular (resp. 0,n-regular) if ∀ℏ∈S∃κ∈S such that ℏ=ℏmκresp.ℏ=κℏn.Lemma 9.
InS, the following assertions hold:(1)
Inm,0-regular semigroup S, ℱℋ^∈℘ℋm,0^⇒ℱℋ^∈℘ℋR^(2)
In0,n-regular semigroup S, ℱℋ^∈℘ℋ0,n^⇒ℱℋ^∈℘ℋL^Proof.
Letℱℋ^∈℘ℋm,0^ and ℏ,κ∈S. Since S is m,0-regular, ∃ℓ∈S such that ℏκ=ℏmℓκ. Therefore, we have(24)ℱℋ^ℏκ=ℱℋ^ℏmℓκ=ℱℋ^ℏmℓκ⊇ℱℋ^ℏ.Hence,ℱℋ^∈℘ℋR^.(2) Similar to the proof of (1).Lemma 10.
Let∅≠Ω⊆S. Then, Ω∈ℐm,0 (resp. Ω∈ℐ0,n) ⇔ HFS χΩℋ^∈℘ℋm,0^ (resp. χΩℋ^∈℘ℋ0,n^).Proof.
(⇒) Let r1,r2,…,rm,z∈S. If xi∉Ω for any i∈1,2,…,m, then(25)χΩℋ^r1r2…rmz⊇χΩℋ^r1∩χΩℋ^r2∩⋯∩χΩℋ^rm.Ifri∈Ω for each i∈1,2,…,m, then r1r2⋯rmz∈ΩmS⊆Ω. Therefore,(26)χΩℋ^r1r2…rmz=1⊇χΩℋ^r1∩χΩℋ^r2∩⋯∩χΩℋ^rm.Hence,Ω∈ℐm,0.(⇐) Suppose that x,z∈S. If x∈Ω, then χΩℋ^xmz⊇χΩℋ^x=1 implies χΩℋ^xmz=1. Therefore, xmz∈Ω. Thus, ΩmS⊆Ω.Theorem 5.
LetGℋ^ be the HFS of S. Then, Gℋ^T∈ℐm,0 (resp. Gℋ^T∈ℐm,0) ∀T∈P0,1, provided Gℋ^T≠∅⇔Gℋ^∈℘ℋm,0^ (resp. Gℋ^∈℘ℋ0,n^).Proof.
(⇒) Suppose that z∈S and r1,r2,…,rm∈GTℋ^,whereT∈P0,1. Then, Gr1ℋ^⊇T,Gr2ℋ^⊇T,…,Grmℋ^⊇T. By Definition 4, Gr1r2⋯rmzℋ^⊇Gr1ℋ^∩Gr2ℋ^∩⋯∩Grmℋ^=T. Therefore, r1r2⋯rmz∈Gℋ^T. Hence, Gℋ^T∈ℐm,n.(⇐) Let Gℋ^T∈ℐm,0∀T∈P0,1 and x,y∈S. Suppose, to the contrary, that Gr1ℋ^∩Gr2ℋ^∩⋯∩Grmℋ^⊃Gr1r2⋯rmzℋ^. Then, there exists C∈P0,1 such that Gr1ℋ^∩Gr2ℋ^∩⋯∩Grmℋ^⊇C⊃Gr1r2⋯rmzℋ^. This implies that r1,r2,…,rm,z,∈Gℋ^C, but r1r2⋯rmz∉Gℋ^C, a contradiction. Thus, Gr1r2⋯rmzℋ^⊇Gr1ℋ^∩Gr2ℋ^∩⋯∩Grmℋ^ for all r1,r2,…,rm,z∈S. Hence, by Definition 4, Gℋ^ is a hesitant fuzzy m,0-ideal of S.Theorem 6.
Letℱℋ^ be any HFSS of S. Then, ℱℋ^∈℘ℋm,0^ (resp. ℱℋ^∈℘ℋ0,n^) ⇔ℱℋ^m°ℐSℋ^⊆ℱℋ^ (resp. ℐSℋ^°ℱℋ^n⊆ℱℋ^).Proof.
On the similar lines to the proof of Theorem2.Lemma 11.
IfS is an m,n-regular semigroup, then ℱℋ^°ℱℋ^=ℱℋ^∀ℱℋ^∈℘ℋm,0^ and Gℋ^°Gℋ^=Gℋ^∀Gℋ^∈℘ℋ0,n^.Proof.
LetS be an m,n-regular semigroup and ℱℋ^∈℘ℋm,0^. Then, ℱℋ^°ℱℋ^⊆ℱℋ^. As S is m,n-regular, we have(27)ℱℋ^⊆ℱℋ^m°ℐSℋ^°ℱℋ^n=ℱℋ^m°ℐSℋ^°ℱℋ^n−1°ℱℋ^⊆ℱℋ^m°ℐSℋ^°ℱℋ^n−1°ℱℋ^m°ℐSℋ^°ℱℋ^n⊆ℱℋ^m°ℐSℋ^°ℱℋ^m°ℐSℋ^⊆ℱℋ^°ℱℋ^,and so, we obtain ℱℋ^⊆ℱℋ^°ℱℋ^. Hence, ℱℋ^=ℱℋ^°ℱℋ^. Similarly, we may prove that Gℋ^°Gℋ^=Gℋ^∀Gℋ^∈℘ℋ0,n^.Theorem 7.
The following statements hold inS:(1)
S is m,0-regular ⇔ℱℋ^⊆ℱℋ^m°ℐSℋ^ for each HFS ℱℋ^ of S(2)
S is 0,n-regular ⇔ℱℋ^⊆ℐSℋ^°ℱℋ^n for each HFS ℱℋ^ of SProof.
(1)
(⇒) Take any ℏ∈S. Then, ∃κ∈S such that ℏ=ℏmκ. Now, we have(28)ℱℋ^m°ℐSℋ^ℏ=∪ℏ=rsℱℋ^mr∩ℐSℋ^s⊇ℱℋ^mℏm∩ℐSℋ^κ=ℱℋ^mℏm=ℱℋ^ℏ.Therefore,ℱℋ^⊆ℱℋ^m°ℐSℋ^.(⇐) Let ℏ∈S. Since χℏℋ^ is the HFS of S, by hypothesis, χℏℋ^⊆χℏℋ^m°ℐSℋ^°χℏℋ^n=ℋℏmSℏn. So, ℏ∈ℏmSℏn, and hence, S is m,n-regular.(2)
Similar to the proof of (1).Theorem 8.
The following assertions are true inS:(1)
S is m,0-regular ⇔ℱℋ^=ℱℋ^m°ℐSℋ^∀ℱℋ^∈℘ℋm,0^(2)
S is 0,n-regular ⇔Gℋ^=ℐSℋ^°Gℋ^n∀Gℋ^∈℘ℋ0,n^Proof.
(1)
(⇒) Let ℱℋ^∈℘ℋm,0^. Then, by hypothesis and Theorems 7 and 6, we have ℱℋ^⊆ℱℋ^m°ℐSℋ^ and ℱℋ^m°ℐSℋ^⊆ℱℋ^. Hence, ℱℋ^=ℱℋ^m°ℐSℋ^.(⇐) Let R be any m,0-ideal of S, and take a∈R. Then, by hypothesis, we have ℋRm°ℐSℋ^a=ℋRa=0,1 implies ℋRm°ℐSℋ^a=0,1. Therefore, there exist elements x,y in S with a=xy such that ℋRmx=0,1 and ℐSℋ^y=0,1. As we have ℋRmx=0,1, ℋRmx=0,1, and it follows that x∈Rm. Since a=xy, therefore, a=xy∈RmS. Thus, R⊆RmS. Since R is m,0-ideal of S, RmS⊆R. Therefore, R=RmS. Hence, by Theorem 1 of [44], S is m,0-regular.(2)
Similar to the proof of (1).Theorem 9.
A semigroupS is m,n-regular ⇔ℱℋ^∩Gℋ^=ℱℋ^m°Gℋ^∩ℱℋ^°Gℋ^n∀ℱℋ^∈℘ℋm,0^ and ∀Gℋ^∈℘ℋ0,n^.Proof.
(⇒) Let ℱℋ^∈℘ℋm,0^ and Gℋ^∈℘ℋ0,n^. As S is m,n-regular, we haveℱℋ^∩Gℋ^⊆ℱℋ^∩Gℋ^m°χℋ^°ℱℋ^∩Gℋ^n⊆ℱℋ^m°χℋ^°Gℋ^n⊆ℱℋ^m°Gℋ^,and so, ℱℋ^∩Gℋ^⊆ℱℋ^m°Gℋ^. Similarly, ℱℋ^∩Gℋ^⊆ℱℋ^°Gℋ^n. Thus, ℱℋ^∩Gℋ^⊆ℱℋ^m°Gℋ^∩ℱℋ^°Gℋ^n. As ℱℋ^∈℘ℋm,0^ and Gℋ^∈℘ℋ0,n^, ℱℋ^m°Gℋ^∩ℱℋ^°Gℋ^n⊆ℱℋ^∩Gℋ^. Therefore, ℱℋ^∩Gℋ^=ℱℋ^m°Gℋ^∩ℱℋ^°Gℋ^n.(⇐) Let a∈R∩L for R∈ℐm,0 and L∈ℐ0,n. By Lemma 10, χRℋ^∈℘ℋm,0^ and χLℋ^∈℘ℋ0,n^. Therefore, by hypothesis,(29)χRℋ^∩χLℋ^a=χRℋ^m°χLℋ^a∩χRℋ^°χLℋ^na=χRmℋ^°χLℋ^a∩χRℋ^°χLnℋ^a.SinceℋRa=0,1 and ℋLa, χRℋ^∩χLℋ^a=0,1, χRmℋ^°χLℋ^a=0,1 and χRℋ^°χLnℋ^a=0,1. This implies that there exist x,y,u,andv in S with a=xyanda=uv such that χRmℋ^x=0,1,χLnℋ^y=0,1, and χRℋ^u=0,1,χLnℋ^v=0,1, and it follows that x∈Rm,y∈L and u∈R,v∈Ln. As a=xy and a=uv, a=xy∈RmL and a=uv∈RLn imply a∈RmL∩RLn. Thus, we obtain R∩L⊆RmL∩RLn. Also, RmL∩RLn⊆R∩L. Therefore, R∩L=RmL∩RLn. Hence, by Theorem 3 of [44], S is m,n-regular.Proposition 1.
Letℱℋ^∈℘ℋm,0^ and Gℋ^∈℘ℋ0,n^. If ℱℋ^°Gℋ^=Gℋ^°ℱℋ^, then the product ℱℋ^°Gℋ^∈℘ℋm,n^.Proof.
Letℱℋ^°Gℋ^=Gℋ^°ℱℋ^. Then, we have(30)ℱℋ^°Gℋ^°ℱℋ^°Gℋ^=ℱℋ^°Gℋ^°Gℋ^°ℱℋ^⊆ℱℋ^°Gℋ^°ℱℋ^°Gℋ^⊆ℱℋ^°ℱℋ^°Gℋ^⊆ℱℋ^°Gℋ^.
Therefore,ℱℋ^°Gℋ^ is a HFSS of S. Also, we have(31)ℱℋ^°Gℋ^m°χℋ^°ℱℋ^°Gℋ^n=ℱℋ^m°Gℋ^m°χℋ^°ℱℋ^n°Gℋ^n⊆ℱℋ^m°χℋ^°Gℋ^n°ℱℋ^n⊆ℱℋ^m°Gℋ^°ℱℋ^n⊆ℱℋ^°Gℋ^°ℱℋ^⊆ℱℋ^°Gℋ^,as required.Lemma 12.
Letℱℋ^ be a HFS of S. Then, ∪i=1mℱℋ^∪ℱℋ^m°ℐSℋ^∈℘ℋm,0^resp.∪i=1nℱℋ^∪ℐSℋ^°ℱℋ^n∈℘ℋ0,n^.Proof.
Straightforward.Lemma 13.
IfS is m,n-regular, then ∀Jℋ^∈ℐℋ^m,n there exist ℱℋ^∈ℐℋ^m,0 and Gℋ^∈ℐℋ^0,n such that Jℋ^=ℱℋ^°Gℋ^.Proof.
LetJℋ^∈ℐℋ^m,n. Then, Jℋ^m°ℐSℋ^°Jℋ^n⊆Jℋ^. As S is m,n-regular, Jℋ^⊆Jℋ^m°ℐSℋ^°Jℋ^n. Therefore, Jℋ^=Jℋ^m°ℐSℋ^°Jℋ^n. Let ℱℋ^=∪i=1mJℋ^i∪Jℋ^m°ℐSℋ^ and Gℋ^=∪i=1nJℋ^i∪ℐSℋ^°Jℋ^n. By Lemma 12, ℱℋ^∈℘ℋm,0^ and Gℋ^∈℘ℋ0,n^. As S is m,n-regular, ℱℋ^=∪i=1mJℋ^i∪Jℋ^m°ℐSℋ^=Jℋ^m°ℐSℋ^ and Gℋ^=∪i=1nJℋ^i∪ℐSℋ^°Jℋ^n=ℐSℋ^°Jℋ^n. Thus,(32)ℱℋ^°Gℋ^=Jℋ^m°ℐSℋ^°ℐSℋ^°Jℋ^n⊆Jℋ^m°ℐSℋ^°Jℋ^n=Jℋ^Jℋ^=Jℋ^m°ℐSℋ^°Jℋ^n=Jℋ^m°ℐSℋ^°Jℋ^nmχℋ^°Jℋ^n=Jℋ^m°χℋ^°Jℋ^n°Jℋ^m°χℋ^°Jℋ^n…Jℋ^m°χℋ^°Jℋ^n︸m−times°χℋ^°Jℋ^n⊆Jℋ^m°χℋ^°Jℋ^n°Jℋ^m°χℋ^°Jℋ^n°⋯°Jℋ^m°χℋ^°Jℋ^n°Jℋ^n=Jℋ^m°χℋ^°Jℋ^n°Jℋ^m°χℋ^°Jℋ^n°⋯°Jℋ^m°χℋ^°Jℋ^n°χℋ^°Jℋ^n⊆Jℋ^m°χℋ^°χℋ^°ℋn=ℱℋ^°Gℋ^,as required.Lemma 14.
IfS is m,n-regular, then ∀ℱℋ^∈℘ℋm,0^, and for each HFS Gℋ^ of S, ℱℋ^°Gℋ^∈℘ℋm,n^.Proof.
Letℱℋ^∈℘ℋm,0^ and Gℋ^ be the HFS of S. We have(33)ℱℋ^°Gℋ^m°ℐSℋ^°ℱℋ^°Gℋ^n=ℱℋ^°Gℋ^°ℱℋ^°Gℋ^°⋯°ℱℋ^°Gℋ^︸m−times°ℐSℋ^°ℱℋ^°Gℋ^°ℱℋ^°Gℋ^°⋯°ℱℋ^°Gℋ^︸n−times=ℱℋ^°Gℋ^°ℱℋ^°Gℋ^°ℱℋ^°Gℋ^°⋯°ℱℋ^°Gℋ^︸m−1−times°ℐSℋ^°ℱℋ^°Gℋ^°ℱℋ^°Gℋ^°⋯°ℱℋ^°Gℋ^︸n−1−times°ℱℋ^°Gℋ^⊆ℱℋ^°Gℋ^°ℐSℋ^°ℐSℋ^°ℐSℋ^°ℱℋ^°Gℋ^⊆ℱℋ^°ℐSℋ^°Gℋ^⊆ℱℋ^m°ℐSℋ^°Gℋ^by Lemma5⊆ℱℋ^°Gℋ^.
Therefore,ℱℋ^°Gℋ^∈℘ℋm,n^.
By Lemmas13 and 14, we have the following.Corollary 1.
IfS is m,n-regular, then Jℋ^∈℘ℋm,n^⇔ there exist ℱℋ^∈℘ℋm,0^ and Gℋ^∈℘ℋ0,n^ such that Jℋ^=ℱℋ^°Gℋ^.


## 5. Conclusion

The principal objective of this paper is to establish the notions of the hesitant fuzzy - ideal, hesitant fuzzym,0-ideal, and hesitant fuzzy 0,n-ideal and to improve the understanding of various semigroup classes through the use of these notions. In particular, if we take m=1=n in the hesitant fuzzy m,n-ideal, hesitant fuzzy m,0-ideal, and hesitant fuzzy 0,n-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 m=1=n 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.



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