On inf -Hesitant Fuzzy Γ -Ideals of Γ -Semigroups

The notions of an inf-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.


Introduction
e 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. e 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][6][7][8][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][12][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. ey 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 this paper, the notions of an inf-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 Γsemigroup 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 Γsemigroup are given in terms of inf-hesitant and (sup, inf)-hesitant fuzzy Γ-ideals.

Preliminaries
We will introduce some definitions and results that are important for study in this paper.
First, we recall the definition of Γ-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, c, v)↦ucv satisfying the identity (ucv)δw � uc(vδw) for all u, v, w ∈ G and c, δ ∈ Γ. From now on throughout this paper, G is represented as a Γsemigroup and X a nonempty set unless otherwise specified. For nonempty subsets U and V of G, let UΓV � ucv|u ∈ U, v ∈ V, c ∈ Γ . By a Γ-ideal (ΓId) of G, we mean a nonempty subset V of G such that GΓV⊆V and VΓG⊆V. en, a nonempty subset V of G is a ΓId of G if and only if ucv, vcu ∈ V for all u ∈ G, v ∈ V, and c ∈ Γ.
A fuzzy subset (FS) [1] of X is a function from X into the unit segment of the real 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 ≤ ϕ( for all x ∈ X. en, ((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 fuzzy Γ-ideal (IFΓId) [36] of G if the following two conditions hold: By an interval number a, we mean an interval [a − , a + ], where 0 ≤ a − ≤ a + ≤ 1.
e set of all interval numbers is denoted by D[0, 1]. Especially, we denoted 1: � [1, 1] and 0: , define the operations ≺ , �, ≺ and rmax in case of two elements in D[0, 1] as follows: A hesitant fuzzy set (HFS) [3,4] on X in terms of a function ψ is that when applied to X returns a subset of denotes the set of all subsets of [0, 1]. Let HFS(X) be the set of all HFSs on X, that is, HFS(X) � ψ|ψ: X ⟶ P([0, 1]) and let HFS * (X) � ψ ∈ HFS(X)|ψ(x) ≠ ∅ for all x ∈ X . en, For ψ ∈ HFS(X) and ∇∈ P([0, 1]), we define the element SUP∇ of [0, 1], the subset [ψ; ∇] SUP of X, the fuzzy subset F ψ of X, and the hesitant fuzzy set H SUP (ψ; ∇) on X [15,16] by 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.

Theorem 2 (see [38]). A HFS ψ on G is a sup-HFΓId of G if
For every HFS ψ on X and every element t of [0, 1], the set is called a sup-upper t-level subset [13,38] of ψ.

Theorem 3 (see [38]). A HFS ψ on G is a sup-HFΓId of G if and only if U SUP (ψ; t) is either empty or a ΓId of G for all
Δ otherwise.  [38] gave conditions for a nonempty subset Y of G to be a ΓId by using the CIvFS CI Y , the CHFS CH Y , and χ (Δ,∇) Y as the following theorem.
Theorem 4 (see [38]). For a nonempty subset Y of G, the following are equivalent:

inf-Hesitant Fuzzy Γ-Ideals
Note that for all x ∈ X and for all ω ∈ IvFS(X), we have . Now, we introduce the notion of an inf-hesitant fuzzy Γ-ideal of a Γsemigroup in the following definition. (1) Define a HFS ψ on G by

Definition 2. A HFS ψ on G is called an inf-hesitant fuzzy
for all u ∈ G. en, ψ is an inf-HFΓId of G but not a sup-HFΓId of G because (2) Define a HFS ψ on G by (10)

Advances in Fuzzy Systems
By Example 1 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.

□
In the following example, it is shown that the converse of Lemma 3 is not generally true.

(13)
By Lemma 3 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 HFS ψ on X, define the FS F ψ of X by for all x ∈ X. Let IC(ψ) 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 * ∈ IC(ψ),

□
In the following theorem, we give conditions for a HFS of a Γ-semigroup to be an inf-HFΓId via IFSs.