A REMARK ON HESITANT FUZZY IDEALS IN ORDERED SEMIGROUPS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. We investigate the concept of hesitant fuzzy sets. This concept is a generalized notion of fuzzy sets. The concept of fuzzy sets can be applied to the study of ordered semigroups. It is well-known that in regular ordered semigroups, any fuzzy bi-ideal and fuzzy quasi-ideal coincide. In this paper, we illustrate that this fact does not hold for hesitant fuzzy sets. Furthermore, we provide an example illustrating that in a regular ordered semigroup, there is a hesitant fuzzy bi-ideal which is not a hesitant fuzzy quasi-ideal. Moreover, we present a class of hesitant fuzzy sets in which hesitant fuzzy bi-ideals and hesitant fuzzy quasi-ideals coincide.


INTRODUCTION
The concept of fuzzy sets was first introduced by Zadeh as a generalization of vague sets (see [20]). Fuzzy sets are widely utilized in many branches of research, especially in decision making problems (see [5,21]). Moreover, fuzzy sets can be applied to investigate algebraic properties of semigroups and their generalizations (see [4,7,8,9,11,12,18]).
There are many concepts extending the application of fuzzy sets; among them is the concept of hesitant fuzzy sets established by Torra. The notion was introduced for situations that the degree of membership is difficult to specify (see [15]). Therefore, hesitant fuzzy sets are suitable tools to deal with practical problems in decision making (see [16,17]). Moreover, the concept of hesitant fuzzy sets can also be used to investigate properties of algebras. In 2016, Ahn, Lee and Jun extended the notions of ideals in ordered semigroups, relating it to the notions of hesitant fuzzy ideals. Based on their results, Ahn, Lee and Jun introduced the notions of hesitant fuzzy left, right, quasi-and bi-ideals. They also studied the interconnections between these kinds of hesitant fuzzy ideals in ordered semigroups (see [3]). Later on, many researchers conducted studies into algebraic systems in terms of hesitant fuzzy sets in various directions (see [1,2,6,13,14,19]).
In the investigation of fuzzy ideals in ordered semigroups, every fuzzy quasi-ideal of an ordered semigroup is a fuzzy bi-ideal. The converse of this statement is true if an ordered semigroup is regular (see [10]). This assertion was explored and proven in terms of hesitant fuzzy sets by Ahn, Lee and Jun. They illustrated that every hesitant quasi-ideal of an ordered semigroup is a hesitant bi-ideal. Moreover, the authors illustrated that if an ordered semigroup is regular, then the concepts of hesitant quasi-ideal and hesitant bi-ideal coincide (see [3]).
In this paper, we illustrate using an example that in a regular ordered semigroup, a hesitant fuzzy bi-ideal may not always be a hesitant fuzzy quasi-ideal. Finally, we establish a particular class of hesitant fuzzy sets, the properties of which that every hesitant fuzzy bi-ideal is a hesitant fuzzy quasi-ideal in regular ordered semigroups.

PRELIMINARIES
We provide brief concepts of ordered semigroups and hesitant fuzzy ordered semigroups in this section.
2.1. Ordered semigroups. The structure S; •, ≤ consisting of a nonempty set S, a binary operation • defined on S and a binary relation ≤ defined on S is called an ordered semigroup if (1) S; • is a semigroup, that is, • is associative, Throughout this paper, we will denote x • y by xy for any x, y ∈ S. We simplify ordered semigroup S; •, ≤ as S.
We observe that any semigroup can be regarded as an ordered semigroup by defining ≤ := ∆, where ∆ is the equality relation on the underlying set of such semigroup.
Let S be an ordered semigroup, and A, B ⊆ S. We define the sets We denote the set of all left ideals, right ideals, quasi-ideals, and bi-ideals of an ordered semigroup S by L(S), R(S), Q(S), and B(S), respectively. Thus, it is illustrated that L(S) ⊆ Q(S), R(S) ⊆ Q(S) and Q(S) ⊆ B(S).
In general, the sets Q(S) and B(S) may not be equal. But Q(S) = B(S) if S is a regular ordered semigroup. A regular ordered semigroup will mean that an ordered semigroup such that for any a ∈ S, we have a ≤ axa for some instances x ∈ S.  We denote the set of all fuzzy sets on X and hesitant fuzzy sets on X by FS(X) and HFS(X), respectively.
Remark 2.1. The concept of hesitant fuzzy sets is a generalization of fuzzy sets in the sense that a fuzzy set can be regarded as a hesitant fuzzy set. In fact, there is a one-to-one function Let S be an ordered semigroup, and x ∈ S. We denote the set Let S be an ordered semigroup, and f , g ∈ FS(S). We define a binary relation ⊆, and binary operations ∩, ∪ and • on the set FS(S) as follows: for all x ∈ S.
Let S be an ordered semigroup. A fuzzy set f of S, such that for any x, y ∈ S, we have Let S be an ordered semigroup. A hesitant fuzzy set H of S satisfying x ≤ y implies that H x ⊇ H y for all x, y ∈ S which is said to be (1) a hesitant fuzzy left (resp., right) ideal of S if H xy ⊇ H y (resp., H xy ⊇ H x ) for all x, y ∈ S, respectively. More information about hesitant fuzzy ideals in ordered semigroups can be found in [3].
The following theorem was proven by Ahn, Lee and Jun in 2016.

RESULTS
Firstly, we give an example which illustrates that Theorem 2.3(4) is not true.
for any x ∈ S. We show that H is a hesitant fuzzy bi-ideal of S. Now, the question becomes when does a hesitant fuzzy bi-ideal a hesitant fuzzy quasi-ideal of an ordered semigroup? Before we address this question, we introduce a notion which is important for this answer.
Let S be an ordered semigroup, and H ∈ HFS(S). We call H a hesitant fuzzy set of S with chain-valued if either H x ⊆ H y , or H x ⊇ H y for any x, y ∈ S. We denote the set of all hesitant fuzzy sets of S with chain-valued by c-HFS(S).
Our main theorem shows that Theorem 2.3(4) is true only in a particular class of hesitant fuzzy sets of S. Theorem 3.2. Let S be a regular ordered semigroup, and let H ∈ c-HFS(S). In this case, the following statements are equivalent.
(1) H is a hesitant fuzzy quasi-ideal of S.
(2) H is a hesitant fuzzy bi-ideal of S.