# Oninf-Hesitant Fuzzy Γ-Ideals of Γ-Semigroups

Authors: Pongpun Julatha; Aiyared Iampan
Journal: Advances in Fuzzy Systems
 (2022)

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## Abstract

The notions of aninf-hesitant fuzzy Γ-ideal and a sup,inf-hesitant fuzzy Γ-ideal, which are a generalization of an interval-valued fuzzy Γ-ideal, of a Γ-semigroup are introduced and some properties are investigated. Characterizations of the notions are provided in terms of sets, fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. Furthermore, characterizations of a Γ-ideal of a Γ-semigroup are given in terms of inf-hesitant and sup,inf-hesitant fuzzy Γ-ideals.


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## Body

## 1. Introduction

The notion of a fuzzy set, proposed by Zadeh [1], has provided a useful mathematical tool and method for describing the behavior of complex and ill-defined systems. The notion has huge applications in decision making, artificial intelligence, automata theory, control engineering, finite state machine, expert, graph theory, robotics, and many branches of pure and applied mathematics (cf. [2]). Nevertheless, there are limitations for using the notion to deal with vague and imprecise information when different sources of vagueness appear simultaneously. In order to overcome such limitations, Torra and Narukawa [3, 4] proposed an extension of the notion so-called a hesitant fuzzy set which is a function from a reference set to a power set of the unit interval. Hesitant fuzzy set theory has been applied to several practical problems, primarily in the area of decision making (see [5–9]) and different algebraic structures; for example, Jun and Ahn [10] introduced hesitant fuzzy subalgebras and hesitant fuzzy ideals of BCK/BCI-algebras and investigated related properties. Mosrijai et al. [11–13] studied hesitant fuzzy sets on UP-algebras. Kim et al. [14] studied the concepts and properties of a hesitant fuzzy subgroupoid (left ideal, right ideal, and ideal) of a groupoid, a hesitant fuzzy subgroup (normal subgroup and quotient subgroup) of a group, and a hesitant fuzzy subring (left ideal, right ideal, and ideal) of a ring. Jittburus and Julatha [15] proposed the concepts of a sup-hesitant fuzzy ideal of a semigroup and its sup-hesitant fuzzy translations and sup-hesitant fuzzy extensions. They showed that the sup-hesitant fuzzy ideal is a general concept of a hesitant fuzzy ideal and an interval-valued fuzzy ideal and gave its characterizations in terms of sets, fuzzy sets, hesitant fuzzy sets, and interval-valued fuzzy sets. Julatha and Iampan [16] introduced sup-types of hesitant fuzzy sets based on ideal theory of ternary semigroups and examined their properties via a fuzzy set, an interval-valued fuzzy set, and a hesitant fuzzy set.In 1981, Sen [17] introduced the concept and notion of the Γ-semigroup as a generalization of the plain semigroup and ternary semigroup. Many classical notions and results of (ternary) semigroups have been extended and generalized to Γ-semigroups, by many mathematicians, for instance, Siripitukdet and Iampan [18, 19], Siripitukdet and Julatha [20], Dutta and Adhikari [21, 22], Saha and Sen [23–25], Hila [26, 27], and Chinram [28, 29]. Simuen, Iampan, Chinram, Sardar, Majumder, Dutta, and Davvaz [30–35] studied theory of Γ-semigroups via fuzzy subsets. Uckun et al. [36] studied theory of Γ-semigroup via intuitionistic fuzzy subsets. Abbasi et al. [37] introduced hesitant fuzzy left (resp., right, bi-, interior, and two-sided) Γ-ideals of Γ-semigroups and characterized simple Γ-semigroups by hesitant fuzzy sets. Julatha and Iampan [38] introduced a sup-hesitant fuzzy Γ-ideal, which is a general concept of an interval-valued fuzzy Γ-ideal and a hesitant fuzzy Γ-ideal, of a Γ-semigroup and studied its properties via level sets, fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets.In this paper, the notions of aninf-hesitant fuzzy Γ-ideal and a sup,inf-hesitant fuzzy Γ-ideal, which are a general notion of an interval-valued fuzzy Γ-ideal, of a Γ-semigroup are introduced and their properties are investigated. Equivalent conditions for a hesitant fuzzy set to be an inf-hesitant fuzzy Γ-ideal and a sup,inf-hesitant fuzzy Γ-ideal are provided in terms of sets, fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, and hesitant fuzzy sets. We show that every interval-valued fuzzy set on a Γ-semigroup is an interval-valued fuzzy Γ-ideal if and only if it is a sup,inf-hesitant fuzzy Γ-ideal. Furthermore, characterizations of a Γ-ideal of a Γ-semigroup are given in terms of inf-hesitant and sup,inf-hesitant fuzzy Γ-ideals.


## 2. Preliminaries

We will introduce some definitions and results that are important for study in this paper.First, we recall the definition ofΓ-semigroups which is defined by Sen and Saha [25]. By a Γ-semigroup, we mean a nonempty set G with a nonempty set Γ and a mapping G×Γ×G⟶G, written as u,γ,v↦uγv satisfying the identity uγvδw=uγvδw for all u,v,w∈G and γ,δ∈Γ. From now on throughout this paper, G is represented as a Γ-semigroup and X a nonempty set unless otherwise specified. For nonempty subsets U and V of G, let UΓV=uγv|u∈U,v∈V,γ∈Γ. By a Γ-idealΓId of G, we mean a nonempty subset V of G such that GΓV⊆V and VΓG⊆V. Then, a nonempty subset V of G is a ΓId of G if and only if uγv,vγu∈V for all u∈G, v∈V, and γ∈Γ.Afuzzy subset (FS) [1] of X is a function from X into the unit segment of the real line 0,1. A FS ϕ of G is called a fuzzyΓ-idealFΓId of G if(1)maxϕu,ϕv≤ϕuγv,for allu,v∈Gandγ∈Γ.Anintuitionistic fuzzy set (IFS) A [39] in X is an object having the form A=x,ϕx,φx|x∈X, where the functions ϕ:X⟶0,1 and φ:X⟶0,1 define the degree of membership and the degree of nonmembership, respectively, and 0≤ϕx+φx≤1 for all x∈X. An IFS A=x,ϕx,φx|x∈X in X can be identified to an ordered pair ϕ,φ in 0,1X×0,1X. For a FS ϕ of X, we define a FS ϕ/2 by ϕ/2x=ϕx/2 for all x∈X. Then, ϕ/2,φ/2 is an IFS in X for all FSs ϕ and φ of X. An IFS ϕ,φ in G is called an intuitionistic fuzzyΓ-idealIFΓId [36] of G if the following two conditions hold:(i)
IFΓId1ϕuγv≥maxϕu,ϕv for all u,v∈G and γ∈Γ(ii)
IFΓId2φuγv≤minφu,φv for all u,v∈G and γ∈ΓBy an interval numbera¯, we mean an interval a−,a+, where 0≤a−≤a+≤1. The set of all interval numbers is denoted by D0,1. Especially, we denoted 1¯:=1,1 and 0¯:=0,0. For two elements a¯=a−,a+ and b¯=b−,b+ in D0,1, define the operations ≺¯,=,≺ and rmax in case of two elements in D0,1 as follows:(i)
a¯≺¯b¯⇔a−≤b− and a+≤b+(ii)
a¯=b¯⇔a−=b− and a+=b+(iii)
a¯≺b¯⇔a¯≺¯b¯ and a¯≠b¯(iv)
rmaxa¯,b¯=maxa−,b−,maxa+,b+Denote the case thata−>b− or a+>b+ by a¯⋠b¯. A function ω˜:X⟶D0,1 is called an interval-valued fuzzy set (IvFS) [40] on X, where ω˜x=ω−x,ω+x for all x∈X and ω− and ω+ are FSs of X such that ω−x≤ω+x for all x∈X. Let IvFS(X) be the set of all IvFSs on X. An IvFS ω˜ on G is called an interval-valued fuzzyΓ-idealIvFΓId of G if(2)rmaxω˜u,ω˜v≺¯ω˜uγv,for allu,v∈Gandγ∈Γ.Then,ω˜ is an IvFΓId of G if and only if ω˜u≺¯ω˜uγv and ω˜v≺¯ω˜uγv for all u,v∈G and γ∈Γ.Ahesitant fuzzy set (HFS) [3, 4] on X in terms of a function ψ^ is that when applied to X returns a subset of 0,1, that is, ψ^:X⟶P0,1, where P0,1 denotes the set of all subsets of 0,1. Let HFSX be the set of all HFSs on X, that is, HFSX=ψ^|ψ^:X⟶P0,1 and let HFS∗X=ψ^∈HFSX|ψ^x≠∅for allx∈X. Then, IvFSX⊆HFS∗X⊆HFSX. A HFS ψ^ on G is called a hesitant fuzzyΓ-idealHFΓId [37] of G if(3)ψ^u∪ψ^v⊆ψ^uγv,for allu,v∈Gandγ∈Γ.Then,ψ^ is a HFΓId of G if and only if ψ^v⊆ψ^uγv∩ψ^vγu for all u,v∈G and γ∈Γ.Forψ^∈HFSX and ∇∈P0,1, we define the element SUP∇ of [0, 1], the subset ψ^;∇SUP of X, the fuzzy subset ℱψ^ of X, and the hesitant fuzzy set ℋSUPψ^;∇ on X [15, 16] by(1)
SUP∇=sup∇if∇≠∅0otherwise(2)
ψ^;∇SUP=x∈X|SUPψ^x≥SUP∇(3)
ℱψ^x=SUPψ^x for all x∈X(4)
ℋSUPψ^;∇x=t∈∇|SUPψ^x≥t for all x∈XWe denoteℋSUPψ^;0,1 by ℋSUPψ^, and then, ℋSUPψ^∈IvFSX.Julatha and Iampan [38] introduced a sup-hesitant fuzzy Γ-ideal, which is a generalization of the concepts of an IvFΓId and a HFΓId, of a Γ-semigroup and studied its properties in terms of FSs, IFSs, HFSs, and IvFSs in the following.Definition 1 (see [38]).
Given∇∈P0,1, a HFS ψ^ on G is called a sup-hesitant fuzzyΓ-ideal ofGrelated to∇ (briefly, ∇-sup-hesitant fuzzy Γ-ideal of G) if the set ψ^;∇SUP is a ΓId of G. We say that ψ^ is a sup-hesitant fuzzyΓ-ideal (sup-HFΓId) of G if ψ^ is a ∇-sup-hesitant fuzzy Γ-ideal of G for all ∇∈P0,1 when ψ^;∇SUP≠∅.Lemma 1 (see [38]).
EveryIvFΓId of G is a sup-HFΓId of G.Lemma 2 (see [38]).
EveryHFΓId of G is a sup-HFΓId of G.Theorem 1 (see [38]).
Forψ^∈HFSG, the following are equivalent:(1)
ψ^ is a sup-HFΓId of G(2)
ℱψ^ is a FΓId of G(3)
ℋSUPψ^ is a HFΓId of G(4)
ℋSUPψ^ is an IvFΓId of G(5)
ℋSUPψ^ is a sup-HFΓId of G(6)
ℋSUPψ^;∇ is a HFΓId of G for all ∇∈P0,1Givenψ^∈HFSX, the HFS ψ^∗, defined by ψ^∗x=1−SUPψ^x for all x∈X, is called the supremum complement [13, 38] of ψ^ on X. Then, SUPψ^∗x=1−SUPψ^x for all x∈X and ℱψ^,ℱψ^∗ is an IFS in X.Theorem 2 (see [38]).
A HFSψ^ on G is a sup-HFΓId of G if and only if ℱψ^,ℱψ^∗ is an IFΓId of G.For every HFSψ^ on X and every element t of 0,1, the set(4)USUPψ^;t=x∈X|SUPψ^x≥t,is called a sup-uppert-level subset [13, 38] of ψ^.Theorem 3 (see [38]).
A HFSψ^ on G is a sup-HFΓId of G if and only if USUPψ^;t is either empty or a ΓId of G for all t∈0,1.For a subsetY of X and two elements Δ,∇ of P0,1, define the characteristic interval-valued fuzzy set (CIvFS) CIY:X⟶D0,1, the characteristic hesitant fuzzy set (CHFS) CHY:X⟶P0,1, and the hesitant fuzzy set χYΔ,∇:X⟶P0,1 by for all x∈X,(5)CIYx=1¯,ifx∈Y,0¯,otherwise,CHYx=0,1,ifx∈Y,∅,otherwise,χYΔ,∇x=∇ifx∈Y,Δotherwise.Then, the HFSχYΔ,∇ is a general concept of the CHFS and CIvFS, that is, χY∅,0,1=CHY and χY0¯,1¯=CIY. Julatha and Iampan [38] gave conditions for a nonempty subset Y of G to be a ΓId by using the CIvFS CIY, the CHFS CHY, and χYΔ,∇ as the following theorem.Theorem 4 (see [38]).
For a nonempty subsetY of G, the following are equivalent:(1)
Y is a ΓId of G(2)
The CIvFSCIY is a sup-HFΓId of G(3)
The CHFSCHY is a sup-HFΓId of G(4)
χYΔ,∇ is a sup-HFΓId of G for all Δ,∇∈P0,1 with SUPΔ<SUP∇


## 3. inf-Hesitant FuzzyΓ-Ideals

For a HFSψ^ on X and an element ∇∈P0,1, define INF∇ and ψ^;∇INF by(6)INF∇=inf∇,if∇≠∅,0,otherwise,ψ^;∇INF=x∈X|INFψ^x≥INF∇.Note that for allx∈X and for all ω˜∈IvFSX, we have INFω˜x=infω˜x=minω˜x=ω−x. Now, we introduce the notion of an inf-hesitant fuzzy Γ-ideal of a Γ-semigroup in the following definition.Definition 2.
A HFSψ^ on G is called an inf-hesitant fuzzy Γ-ideal (inf-HFΓId) of G if the set ψ^;∇INF is a ΓId of G for all ∇∈P0,1 when ψ^;∇INF≠∅.Example 1.
Letℤ− be the set of all negative integers, G=ℤ−∪0, and Γ=2G. Then, G is a Γ-semigroup with respect to usual multiplication.(1)
Define a HFSψ^ on G by(7)ψ^u=0.5,0.6,0.7,ifu=0,0.3,0.8,ifu∈2ℤ−,∅,otherwise,for allu∈G. Then, ψ^ is an inf-HFΓId of G but not a sup-HFΓId of G because(8)SUPψ^−50−6=SUPψ^0=0.7<0.8=maxSUPψ^−5,SUPψ^−6.(2)
Define a HFSψ^ on G by(9)ψ^u=0,1,ifu=0,0.4,0.8,ifu∈2ℤ−,0.5,0.6,0.7,otherwise,for all u∈G. Then, ψ^ is a HFΓId of G but not an inf-HFΓId of G because the nonempty subset ψ^;0.5,0.6INF of G is not a ΓId of G, that is,(10)−1∈ψ^;0.5,0.6INF,−1−2−3=−6∉ψ^;0.5,0.6INF.By Example1 and Lemma 2, we obtain that an inf-HFΓId of G is not a sup-HFΓId and a HFΓId of G and a sup-HFΓId of G is not an inf-HFΓId of G.Lemma 3.
EveryIvFΓId of G is an inf-HFΓId of G.Proof.
Suppose thatω˜ is an IvFΓId of G and ∇∈P0,1 such that ω˜;∇INF is a nonempty set. Let u∈G, v∈ω˜;∇INF, and γ∈Γ. Since ω˜ is an IvFΓId of G, we get ω˜v≺¯ω˜uγv and ω˜v≺¯ω˜vγu. Thus,(11)INF∇≤INFω˜v=ω−v≤minω−uγv,ω−vγu=minINFω˜uγv,INFω˜vγu,which implies that uγv,vγu∈ω˜;∇INF. Hence, ω˜;∇INF is a ΓId of G. Therefore, ω˜ is an inf-HFΓId of G.In the following example, it is shown that the converse of Lemma3 is not generally true.Example 2.
LetG be a Γ-semigroup defined in Example 1. Define an IvFS ω˜ on G by for all u∈G,(12)ω˜u=1¯,ifu=0,0.5,0.7,ifu∈2ℤ−,0.3,0.8,otherwise.
Then,ω˜ is an inf-HFΓId of G but not an IvFΓId of G because(13)rmaxω˜−1,ω˜−3=0.3,0.8⋠0.5,0.7=ω˜−12=ω˜−1−4−3.By Lemma3 and Example 2, we obtain that an inf-HFΓId of a Γ-semigroup G is a general concept of an IvFΓId of G.For every HFSψ^ on X, define the FS ℱψ^ of X by ℱψ^x=INFψ^x for all x∈X. A HFS ϑ^ on X is called an infimum complement of ψ^ on X if INFϑ^x=1−INFψ^x for all x∈X. Let ICψ^ be the set of all infimum complements of ψ^. Define the HFS ψ^∗ by ψ^∗x=1−INFψ^x for all x∈X, and then we have ψ^∗∈ICψ^, ℱψ^∗x=1−INFψ^x, and INFψ^∗∗x=INFψ^x=ℱψ^x for all x∈X. Note that min1−t1,1−t2=1−maxt1,t2 for all t1,t2∈0,1.Lemma 4.
Ifψ^∈HFS∗G is a HFΓId of G, then ϑ^ is an inf-HFΓId of G for all ϑ^∈ICψ^.Proof.
Suppose thatψ^∈HFS∗G is a HFΓId of G and ϑ^∈ICψ^. Let ∇∈P0,1, u∈G, v∈ϑ^;∇INF, and γ∈Γ. Then, ψ^v⊆ψ^uγv and ψ^v⊆ψ^vγu, and since ψ^∈HFS∗G, we get(14)INFψ^v≥maxINFψ^uγv,INFψ^vγu,(15)INF∇≤INFϑ^v=1−INFψ^v≤1−maxINFψ^uγv,INFψ^vγu=min1−INFψ^uγv,1−INFψ^vγu=minINFϑ^uγv,INFϑ^vγu.
Hence, we haveuγv,vγu∈ϑ^;∇INF. Therefore, we obtain that ϑ^;∇INF is a ΓId of G. Consequently, ϑ^ is an inf-HFΓId of G.Lemma 5.
Forψ^∈HFSG, the following are equivalent:(1)
ψ^ is an inf-HFΓId of G(2)
ℱψ^ is a FΓId of G(3)
INFψ^uγv≥maxINFψ^u,INFψ^v for all u,v∈G and γ∈Γ(4)
INFϑ^uγv≤minINFϑ^u,INFϑ^v for all u,v∈G, γ∈Γ, and ϑ^∈ICψ^(5)
INFψ^∗uγv≤minINFψ^∗u,INFψ^∗v for all u,v∈G and γ∈ΓProof.
1⇒3. Let u,v∈G and γ∈Γ. Then, u∈ψ^;ψ^uINF and v∈ψ^;ψ^vINF. By assumption (1), we get uγv∈ψ^;ψ^uINF and uγv∈ψ^;ψ^vINF. Thus, INFψ^uγv≥maxINFψ^u,INFψ^v.
3⇒2 and 4⇒5. They are clear.
2⇒1. Let ∇∈P0,1, u∈G, v∈ψ^;∇INF, and γ∈Γ. By assumption (2), we have(16)minINFψ^uγv,INFψ^vγu=minℱψ^uγv,ℱψ^vγu≥ℱψ^v=INFψ^v≥INF∇.
Then,uγv,vγu∈ψ^;∇INF. Thus, ψ^;∇INF is a ΓId of G. Therefore, we have that ψ^ is an inf-HFΓId of G.
3⇒4. Let u,v∈G, γ∈Γ, and ϑ^∈ICψ^. By assumption (3), we have(17)INFψ^uγv≥maxINFψ^u,INFψ^v,and then,(18)minINFϑ^u,INFϑ^v=min1−INFψ^u,1−INFψ^v=1−maxINFψ^u,INFψ^v≥1−INFψ^uγv=INFϑ^uγv.
5⇒3. Let u,v∈G and γ∈Γ. By using assumption (5), we get INFψ^∗uγv≤minINFψ^∗u,INFψ^∗v and then(19)INFψ^uγv=1−1−INFψ^uγv=1−INFψ^∗uγv≥1−minINFψ^∗u,INFψ^∗v=1−min1−INFψ^u,1−INFψ^v=1−1−maxINFψ^u,INFψ^v=maxINFψ^u,INFψ^v.Theorem 5.
If an IvFSω˜ of G is an IvFΓId of G, then ℱω˜ is a FΓId of G.Proof.
It follows from Lemmas3 and 5.Theorem 6.
Ifψ^∈HFS∗G is a HFΓId of G, then ℱϑ^ is a FΓId of G for all ϑ^∈ICψ^.Proof.
It follows from Lemmas4 and 5.Forψ^∈HFSX and t∈0,1, the sets(20)UINFψ^;t=x∈X|INFψ^x≥t,LINFψ^;t=x∈X|INFψ^x≤t,are called an inf-uppert-level subset and an inf-lowert-level subset of ψ^, respectively.Theorem 7.
A HFSψ^ of G is an inf-HFΓId of G if and only if for all t∈0,1, a nonempty subset UINFψ^;t of G is a ΓId of G.Proof.
Lett∈0,1 and UINFψ^;t≠∅. Choose ∇∈P0,1 such that INF∇=t, and we get ψ^;∇INF=UINFψ^;t. Since ψ^ is an inf-HFΓId of G, we get that UINFψ^;t=ψ^;∇INF is a ΓId of G.
Conversely, let∇∈P0,1 and ψ^;∇INF≠∅. Choose t:=INF∇, and by the assumption, we obtain that ψ^;∇INF=UINFψ^;t is a ΓId of G. Therefore, ψ^ is an inf-HFΓId of G.Corollary 1.
Letω˜ be an IvFΓId of G. Then, for all t∈0,1, a nonempty subset UINFω˜;t of G is a ΓId of G.Proof.
It follows from Lemma3 and Theorem 7.Theorem 8.
Letψ^∈HFSG and ϑ^∈ICψ^. Then, ϑ^ is an inf-HFΓId of G if and only if for all t∈0,1, a nonempty subset LINFψ^;t of G is a ΓId of G.Proof.
Lett∈0,1 and LINFψ^;t≠∅. There exists ∇∈P0,1 such that INF∇=1−t and then ϑ^;∇INF=LINFψ^;t. Since ϑ^ is an inf-HFΓId of G, we obtain that LINFψ^;t=ϑ^;∇INF is an ΓId of G.
Conversely, let∇∈P0,1 be such that ϑ^;∇INF≠∅. Choose t:=1−INF∇, and by the assumption, we obtain that ϑ^;∇INF=LINFψ^;t is a ΓId of G. Hence, ϑ^ is an inf-HFΓId of G.Corollary 2.
Ifψ^∈HFS∗G is a HFΓIdofG, then for all t∈0,1, a nonempty subset LINFψ^;t of G is a ΓId of G.Proof.
It follows from Lemma4 and Theorem 8.In the following theorem, we give conditions for a HFS of aΓ-semigroup to be an inf-HFΓId via IFSs.Theorem 9.
Forψ^∈HFSG, the following are equivalent:(1)
ψ^ is an inf-HFΓId of G(2)
ℱψ^,ℱϑ^ is an IFΓId of G for all ϑ^∈ICψ^(3)
ℱψ^,ℱψ^∗ is an IFΓId of GProof.
It follows from Lemma5.Corollary 3.
If an IvFSω˜ of G is an IvFΓId of G, then ℱω˜,ℱϑ^ is an IFΓId of G for all ϑ^∈ICω˜.Proof.
It follows from Lemma3 and Theorem 9.Corollary 4.
Ifψ^∈HFS∗G is a HFΓId of G, then ℱϑ^,ℱψ^ is an IFΓId of G for all ϑ^∈ICψ^.Proof.
It follows from Lemma4 and Theorem 9.Forψ^∈HFSX and ∇∈P0,1, we define the HFS ℋINFψ^;∇ on X by(21)ℋINFψ^;∇x=t∈∇|INFψ^x≥tfor allx∈X,and we denote ℋINFψ^;0,1 by ℋINFψ^. Then, the following statements hold:(1)
ℋINFψ^;∇x⊆∇ for all x∈X(2)
ℋINFψ^∈IvFSX(3)
0=minℋINFψ^x≤INFℋINFψ^;∇x≤INFψ^x=maxℋINFψ^x for all x∈XIn the following theorem, we give conditions for a HFS of aΓ-semigroup to be an inf-HFΓId in terms of IvFSs and HFSs.Theorem 10.
Forψ^∈HFSG, the following are equivalent:(1)
ψ^ is an inf-HFΓId of G(2)
ℋINFψ^;∇ is a HFΓId of G for all ∇∈P0,1(3)
ℋINFψ^ is a HFΓId of G(4)
ℋINFψ^ is an IvFΓId of GProof.
1⇒2. Let u,v∈G, γ∈Γ, ∇∈P0,1, and t∈ℋINFψ^;∇u∪ℋINFψ^;∇v. Then, t∈∇ and t≤maxINFψ^u,INFψ^v. By assumption (1) and Lemma 5, we get(22)t≤maxINFψ^u,INFψ^v≤INFψ^uγv.
Thust∈ℋINFψ^;∇uγv. Hence, ℋINFψ^;∇u∪ℋINFψ^;∇v⊆ℋINFψ^;∇uγv. Therefore, we have that ℋINFψ^;∇ is a HFΓId of G.
2⇒3. It is clear.
3⇒4. Let u,v∈G and γ∈Γ. Then, INFψ^u∈ℋINFψ^u and INFψ^v∈ℋINFψ^v. By assumption (3), we get INFψ^u,INFψ^v∈ℋINFψ^uγv. Thus,(23)maxℋINFψ^uγv=INFψ^uγv≥maxINFψ^u,INFψ^v=maxmaxℋINFψ^u,maxℋINFψ^v.
SinceminℋINFψ^x=0 for all x∈G, we have minℋINFψ^uγv≥maxminℋINFψ^u,minℋINFψ^v and so(24)rmaxℋINFψ^u,ℋINFψ^v=maxminℋINFψ^u,minℋINFψ^v,maxmaxℋINFψ^u,maxℋINFψ^v⪯minℋINFψ^uγv,maxℋINFψ^uγv=ℋINFψ^uγv.
Therefore,ℋINFψ^ is an IvFΓId of G.
4⇒1. Let u,v∈G and γ∈Γ. By assumption (4), we get rmaxℋINFψ^u,ℋINFψ^v≺¯ℋINFψ^uγv. Then,(25)INFψ^uγv=maxℋINFψ^uγv≥maxmaxℋINFψ^u,maxℋINFψ^v=maxINFψ^u,INFψ^v.
Therefore, it follows from Lemma5 that ψ^ is an inf-HFΓId of G.Corollary 5.
Letω˜ be an IvFΓId of G. Then, the following hold:(1)
ℋINFω˜;∇ is a HFΓId of G for all ∇∈P0,1(2)
ℋINFω˜ is both a HFΓId and an IvFΓId of GProof.
It follows from Lemma3 and Theorem 10.Corollary 6.
Letψ^∈HFS∗G be a HFΓId of G. Then, the following hold:(1)
ℋINFϑ^;∇ is a HFΓId of G for all ∇∈P0,1 and ϑ^∈ICψ^(2)
ℋINFϑ^ is both a HFΓId and an IvFΓId of G for all ϑ^∈ICψ^Proof.
It follows from Lemma4 and Theorem 10.In the following theorem, we give one characterization of aΓId of a Γ-semigroup in terms of a HFS.Theorem 11.
LetY be a nonempty subset of G and Δ,∇∈P0,1 with INFΔ<INF∇. Then, Y is a ΓId of G if and only if χYΔ,∇ is an inf-HFΓId of G.Proof.
Assume thatY is a ΓId of G. Suppose that χYΔ,∇ is not an inf-HFΓId of G. By Lemma 5, there exist u,v∈G and γ∈Γ such that(26)maxINFχYΔ,∇u,INFχYΔ,∇v>INFχYΔ,∇uγv.
Thus,maxINFχYΔ,∇u,INFχYΔ,∇v=INF∇, which implies that u∈Y or v∈Y. Since Y is a ΓId of G, we get uγv∈Y and so(27)maxINFχYΔ,∇u,INFχYΔ,∇v=INF∇=INFχYΔ,∇uγv,which is a contradiction. Therefore, χYΔ,∇ is an inf-HFΓId of G.
Conversely, letu∈Y,v∈G, and γ∈Γ. Then, χYΔ,∇u=∇ and so maxINFχYΔ,∇u,INFχYΔ,∇v=INF∇. Since χYΔ,∇ is an inf-HFΓId of G and Lemma 5, we get INFχYΔ,∇uγv≥INF∇ and INFχYΔ,∇vγu≥INF∇. Thus, uγv,vγu∈Y. Therefore, Y is a ΓId of G.Theorem 12.
A nonempty subsetY of G is a ΓId of G if and only if the CIvFS CIY is an inf-HFΓId of G.Proof.
It follows from Theorem11.Remark 1.
IfY is a subset of G, then the CHFS CHY is an inf-HFΓId of G.Definition 3.
A HFSψ^ on G is called a sup,inf-hesitant fuzzyΓ-ideal (sup,inf-HFΓId) of G if ψ^ is both an inf-HFΓId and a sup-HFΓId of G.In Theorem13, equivalent conditions for a hesitant fuzzy set to be a sup,inf-HFΓId are given in terms of level sets, FSs, IFSs, IvFSs, and HFSs.Theorem 13.
Forψ^∈HFSG, the following are equivalent:(1)
ψ^ is a sup,inf-HFΓId of G(2)
ℱψ^ and ℱψ^ are FΓIds of G(3)
ℋINFψ^ and ℋSUPψ^ are HFΓIds of G(4)
ℋINFψ^ and ℋSUPψ^ are IvFΓIds of G(5)
ℱψ^,ℱψ^∗ is an IFΓId of G(6)
ℱψ^/2,ℱψ^⋆/2 is an IFΓId of G(7)
ℱψ^/2,ℱϑ^/2 is an IFΓId of G for all ϑ^∈ICψ^(8)
ℋINFψ^;∇ and ℋSUPψ^;Δ are HFΓIds of G for all Δ,∇∈P0,1(9)
For allt1,t2∈0,1, nonempty subsets UINFψ^;t1 and USUPψ^;t2 of G are ΓIds of G(10)
SUPψ^uγv≥maxSUPψ^u,SUPψ^v and INFψ^uγv≥maxINFψ^u,INFψ^v for all u,v∈G and γ∈ΓProof.
It follows from Theorems1, 2, 3, 7, and 10 and Lemma 5.Example 3.
Letℤ− be the set of all negative integers and G=Γ=ℤ−. Then, G forms a Γ-semigroup with the usual multiplication. Define a HFS ψ^ on G by ψ^u=u+1/3u,3u+1/3u for all u∈G. Then,(28)SUPψ^uγv=3uγv+13uγv≥max3u+13u,3v+13v=maxSUPψ^u,SUPψ^v,INFψ^uγv=uγv+13uγv≥maxu+13u,v+13v=maxINFψ^u,INFψ^v,for all u,v∈G and γ∈Γ. Hence, it follows from Theorem 13 that ψ^ is a sup,inf-HFΓId of G. Since ψ^ is not an IvFS of G, we get that it is not an IvFΓId of G.Lemma 6.
EveryIvFΓId of G is a sup,inf-HFΓId of G.Proof.
It follows from Lemmas1 and 3.By Example3 and Lemma 6, we see that a sup,inf-HFΓId of G is a general concept of an IvFΓId of G.Lemma 7.
Letω˜ be an IvFS of G. Then, ω˜ is an IvFΓId of G if and only if ω˜ is asup,inf-HFΓId of G.Proof.
It follows from Lemma6.
Conversely, assume thatω˜ is a sup,inf-HFΓId of G. By Theorem 13, we get SUPω˜uγv≥maxSUPω˜u,SUPω˜v and INFω˜uγv≥maxINFω˜u,INFω˜v for all u,v∈G and γ∈Γ. Thus,(29)rmaxω˜u,ω˜v=maxINFω˜u,INFω˜v,maxSUPω˜u,SUPω˜v⪯INFω˜uγv,SUPω˜uγv=ω˜uγv,for all u,v∈G and γ∈Γ. Therefore, ω˜ is an IvFΓId of G.In Theorem14, equivalent conditions for an IvFS to be an IvFΓId are given in terms of level sets, FSs, IFSs, IvFSs, and HFSs.Theorem 14.
Forω˜∈IvFSG, the following are equivalent:(1)
ω˜ is an IvFΓId of G(2)
ω˜ is a sup,inf-HFΓId of G(3)
ℱω˜ and ℱω˜ are FΓIds of G(4)
ℋINFω˜ and ℋSUPω˜ are HFΓIds of G(5)
ℋINFω˜ and ℋSUPω˜ are IvFΓIds of G(6)
ℱω˜,ℱω˜∗ is an IFΓId of G(7)
ℱω˜/2,ℱω˜∗/2 is an IFΓId of G(8)
ℱω˜/2,ℱϑ^/2 is an IFΓId of G for all ϑ^∈ICω˜(9)
ℋINFω˜;∇ and ℋSUPω˜;Δ are HFΓIds of G for all Δ,∇∈P0,1(10)
For allt1,t2∈0,1, nonempty subsets UINFω˜;t1 and USUPω˜;t2 of G are ΓIds of GProof.
It follows from Lemma7 and Theorem 13.Theorem 15.
For a nonempty subsetY of G, the following are equivalent:(1)
Y is a ΓId of G(2)
The CIvFSCIY is a sup,inf-HFΓId of G(3)
The CHFSCHY is a sup,inf-HFΓId of G(4)
χYΔ,∇ is a sup,inf-HFΓId of G for all Δ,∇∈P0,1 with INFΔ<INF∇ and SUPΔ<SUP∇Proof.
It follows from Theorems4, 11, and 12 and Remark 1.


## 4. Conclusions

In this paper, we have introduced the notions of aninf-HFΓId and a sup,inf-HFΓId, which are a generalization of an IvFΓId, of a Γ-semigroup and examined their characterizations in terms of sets, FSs, IFSs, IvFSs, and HFSs. Furthermore, we have discussed the relation between a ΓId and a generalization of the CHFS and CIvFS. From the study results, we found that the following conditions are all equivalent in a Γ-semigroup: a nonempty subset Y is a ΓId, CIY is an inf-HFΓId, CIY is a sup,inf-HFΓId, and CHY is a sup,inf-HFΓId.In the future, we will study aninf-HFΓId and a sup,inf-HFΓId in LA-semigroups and UP-algebras and examine their characterizations in terms of sets, FSs, IFSs, IvFSs, and HFSs.



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