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A Hesitant Fuzzy Set Approach to Ideal Theory in [GAMMA]-Semigroups.

1. Introduction

The concept of a fuzzy set, introduced by Zadeh [1], provides a natural framework for generalizing some of the notions of classical algebraic structures. After the introduction of the concept of fuzzy sets by Zadeh, several researchers conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer science, artificial intelligence, control engineering, expert, robotics, automata theory, finite state machine, graph theory logic, and many branches of pure and applied mathematics (cf. [2]). Fuzzy set theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. But in fuzzy sets theory, there is no means to incorporate the hesitation or uncertainty in the membership degrees and the fuzzy set fails in cases when it has to manage the insufficient understanding of membership degrees. For this, different extensions and generalizations of fuzzy sets have been introduced, such as (i) Atanassov's intuitionistic fuzzy sets (IFS) [3] which give both a membership degree and a nonmembership degree, (ii) type-2 fuzzy sets (T2FS) [4] which let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems and that incorporate uncertainty about the membership function into fuzzy set theory (the value of membership function is given by a fuzzy set), (iii) interval-valued fuzzy sets (IVFS) [4, 5] in which the values of the membership degrees are intervals of numbers instead of the numbers, and thus, i-v fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets, and (iv) fuzzy multisets [6] based on multisets, that is, fuzzy subsets whose elements may occur more than one time (an element of a fuzzy multiset can occur more than once with possibly the same or different membership values). As a new generalization of fuzzy sets, the so-called hesitant fuzzy sets (briefly HFSs) has been introduced by Torra [7-9] as more suitable tools for dealing with group decision-making problems when experts have a hesitation among several possible memberships for an element to a set, rather than a margin of error considered in intuitionistic fuzzy sets (IFSs) or some possibility distribution on the possible values considered in type-2 fuzzy sets (cf. [10]). The HFS maps the membership degree of an element to a set presented as several possible values between zero and one, which can better describe the situations where people have hesitancy in providing their preferences over objects in the process of decision-making and to get the optimal alternative in a decision-making problem with multiple attributes and multiple persons. During the evaluating process in practice, however, these possible memberships may be not only crisp values in [0,1], but also interval values. HFS theory has been applied to different algebraic structures. The structure of semigroups containing hesitant fuzzy ideals was studied by Jun et al. [11-13]. Abbasi et al. [14, 15] applied the notion of HFSs to po-semigroups. Ali et al. [16] studied the notion of HFSs in AG-Groupoids.

In 1981, Sen [17] introduced the concept and notion of the [GAMMA]-semigroup as a generalization of plain semigroup and ternary semigroup. Many classical notions and results of the theory of semigroups have been extended and generalized to [GAMMA]-semigroups, by many mathematicians, for instance, Dutta and Adhikari [18,19], SahaandSen [20-22], and Hila [23-25]. Mustafa et al. [26] have introduced the [GAMMA]-semigroup in terms of intuitionistic fuzzy sets. Sardar, Majumder, Dutta, Davvaz, and Hila [27-30] studied the [GAMMA]-semigroup in terms of fuzzy subsets. Considering all the reasons described above about the problems faced with several generalizations of fuzzy sets in different algebraic structures, which led to the introduction of HFS as a better tool, in this paper, our main aim is to apply the HFS theory to [GAMMA]-semigroups as a more general algebraic structure. We introduce the hesitant fuzzy left (resp., right and two-sided) ideal, hesitant fuzzy bi-ideal, and hesitant fuzzy interior ideal in [GAMMA]-semigroup and study some properties of them. Finally, a characterization of a simple [GAMMA]-semigroup by means of a hesitant fuzzy simple [GAMMA]-semigroup is obtained.

2. Preliminaries

In this section, we introduce some necessary notions on [GAMMA]-semigroups and present some operations on HFSs that will be used throughout the paper.

We first recall the definition of the [GAMMA]-semigroup as a generalization of semigroup and ternary semigroup in another way as follows (cf. [17, 22, 24]).

Definition 1. Let M and [GAMMA] be two nonempty sets. Denote by the letters of the English alphabet the elements of M and with the letters of the Greek alphabet the elements of [GAMMA]. Any map from M x [GAMMA] x M [right arrow] M will be called a [GAMMA]-multiplication in M and denoted by [(*).sub.[GAMMA]]. The result of this multiplication for a, b [member of] M and [alpha] [member of] [GAMMA] is denoted by a[alpha]b. A [GAMMA]-semigroup M is an ordered pair (M, [(*).sub.[GAMMA]]) where M and [GAMMA] are nonempty sets and [(*).sub.[GAMMA]] is a [GAMMA]-multiplication on M which satisfies the following property: [for all](a, b, c, [alpha], [beta]) [member of] [M.sup.3] x [[GAMMA].sup.2], (a[alpha]b)[beta]c = a[alpha](b[beta]c).

This kind of structure is often called one sided [GAMMA]-semigroup or simply [GAMMA]-semigroup.

Example 2. Let M be a semigroup and [GAMMA] be any nonempty set. Define a mapping M x [GAMMA] x M [right arrow] M by a[gamma]b = ab for all a, b [member of] M and [gamma] [member of] [GAMMA]. Then M is a [GAMMA]-semigroup.

Example 3. Let S be the set of all m x n matrices with entries from a field, where m, n are positive integers. Let P(S) be the power set of S. Then it is easy to see that P(S) is not a semigroup under multiplication of matrices because, for A, B [member of] P(S), the product AB is not defined. Let [GAMMA] be the set of n x m matrices with entries from the same field. Then for A, B, C [member of] P(S) and P, Q [member of] [GAMMA], we have APB [member of] P(S), AQB [member of] P(S) and since the matrix multiplication is associative, we get that S is a [GAMMA]-semigroup.

Example 4. Let M be a set of all negative rational numbers. Obviously M is not a semigroup under usual product of rational numbers. Let [GAMMA] = {-1/p : p is prime}. Let a, b, c [member of] M and [alpha] [member of] [GAMMA]. Now if a[alpha]b is equal to the usual product of rational numbers a, [alpha], b, then a[alpha]b [member of] M and (a[alpha]b)[beta]c = a[alpha](b[beta]c). Hence M is a [GAMMA]-semigroup.

Example 5. Let M = {-i, 0, i} and [GAMMA] = M. Then M is a [GAMMA]-semigroup under the multiplication over complex numbers while M is not a semigroup under complex number multiplication.

These examples show that every semigroup is a [GAMMA]-semigroup. Therefore, [GAMMA]-semigroups are a generalization of semigroups.

A nonempty subset K of a [GAMMA]-semigroup M is called a sub-[GAMMA]-semigroup of M if for all a, b [member of] K and [gamma] [member of] [GAMMA], a[gamma]b [member of] K.

Example 6. Let M = [0,1] and [GAMMA] = {1/n | n is a positive integer}. Then M is a [GAMMA]-semigroup under usual multiplication. Let K = [0,1/2]. We have that K is a nonempty subset of M and a[gamma]b [member of] K for all a, b [member of] K and [gamma] [member of] [GAMMA]. Then K is a sub-[GAMMA]-semigroup of M.

Definition 7. The element a of a [GAMMA]-semigroup M is called regular in M if a [member of] a[GAMMA]M[GAMMA]a, where a[GAMMA]M[GAMMA]a = {(a[alpha]b)[beta]a | a, b [member of] M, [alpha], [beta] [member of] [GAMMA]}. M is called regular if and only if every element of M is regular.

The following is the definition of the so-called both-sided [GAMMA]-semigroup [18].

Definition 8. Let M and [GAMMA] be two nonempty sets. Denote by the letters of the English alphabet the elements of M and with the letters of the Greek alphabet the elements of [GAMMA]. Any map from M x [GAMMA] x M [right arrow] M will be called a [GAMMA]-multiplication in M and denoted by [(*).sub.[GAMMA]]. The result of this multiplication for a, b [member of] M and [alpha] [member of] [GAMMA] is denoted by a[alpha]b. Any map from [GAMMA] x M x [GAMMA] [right arrow] [GAMMA] will be called a M-multiplication in [GAMMA] and denoted by [(*).sub.M]. The result of this multiplication for [alpha], [beta] [member of] [GAMMA] and a [member of] M is denoted by [alpha]a[beta]. A both-sided [GAMMA]-semigroup M is an ordered triple (M, [(*).sub.[GAMMA]], [(*).sub.M]) where [(*).sub.[GAMMA]] and [(*).sub.M] satisfy the following property: [for all](a, b, c, [alpha], [beta], [gamma]) [member of] [M.sup.3] x [[GAMMA].sup.3],

(a[alpha]b) [beta]c = a ([alpha]b[beta]) c = a[alpha] (b[beta]c) and

[alpha] (a[beta]b) [gamma] = ([alpha]a[beta]) b[gamma] = [alpha]a ([beta]b[gamma]). (1)

It is clear that a both-sided [GAMMA]-semigroup is always a one sided [GAMMA]-semigroup, while the converse does not hold in general.

Clearly, a both-sided [GAMMA]-semigroup S is regular if and only if a e ala, for all a e S [31].

Throughout this paper unless or otherwise mentioned S stands for one sided [GAMMA]-semigroup and call it simply [GAMMA]-semigroup.

Let S be a reference set. A HFS on S is a function H that when applied to S returns a finite subset of values in [0,1]:

H : S [right arrow] P ([0,1]) (2)

where P([0,1]) denotes the set of all subsets of [0,1].

Let H and G be any two HFSs on S. We define H [??] G if H(a) [subset or equal to] G(a), for all a [member of] S.

Let S be a [GAMMA]-semigroup. For any HFSs H and G in S, the hesitant fuzzy product of H and G is defined to be the HFS [H[??].sub.[GAMMA]]G on S as follows:

[mathematical expression not reproducible] (3)

For any two HFSs H and G on S, the hesitant union H [??] G of H and G is defined to be HFS on S as follows:

[mathematical expression not reproducible] (4)

and the hesitant intersection H [??] G of H and G is defined to be HFS on S as follows:

[mathematical expression not reproducible]. (5)

Let A be any nonempty subset of S. Recall that, we denote by [H.sub.A] the characteristic HFS on S as follows:

[mathematical expression not reproducible] (6)

Let S bea [GAMMA]-semigroup. We define a hesitant fuzzy subset I of S as follows:

I : S [right arrow] (P [0, 1]) | x [right arrow] [I.sub.x] := [0,1]. (7)

3. Main Results

Definition 9. Let S be a [GAMMA]-semigroup and H be a HFS on S. Then H is called

([h.sub.1]) a hesitant fuzzy [GAMMA]-semigroup on S if it satisfies ([for all](x, y, [gamma]) [member of] [S.sup.2] x [GAMMA])(H(x[gamma]y) [contains as member or equal to] H(x) [intersection] H(y));

([h.sub.2]) a hesitant fuzzy left ideal on S if it satisfies ([for all](x, y, [gamma]) [member of] [S.sup.2] x [GAMMA])(H(x[gamma]y) [contains as member or equal to] H(y));

([h.sub.3]) a hesitant fuzzy right ideal on S if it satisfies ([for all](x, y, [gamma]) [member of] [S.sup.2] x [GAMMA])(H(x[gamma]y) [contains as member or equal to] H(x);

([h.sub.4]) a hesitant fuzzy ideal on S if H is both a hesitant fuzzy left ideal and a hesitant fuzzy right ideal on S;

([h.sub.5]) a hesitant fuzzy bi-ideal on S if it satisfies ([for all](x, y, z, [alpha], [beta]) [member of] [S.sup.3] x [[GAMMA].sup.2])(H(x[alpha]y[beta]z) [contains as member or equal to] H(x) [intersection] H(z)).

Example 10. Let S = [M.sub.1x2]([Z.sub.2]) be a set of all 1x2 matrices over the field [Z.sub.2] and [GAMMA] = [M.sub.2x1]([Z.sub.2]) be a set of all 2x1 matrices over the field [Z.sub.2]. Then S is a [GAMMA]-semigroup with respect to usual matrix products a[alpha]b and [alpha]a[beta], for all a, b [member of] S and [alpha], [beta] [member of] [GAMMA].

Define a hesitant fuzzy subset H of S such that

[mathematical expression not reproducible] (8)

Clearly, H is a hesitant fuzzy ideal of S.

Example 11. Let S = [Z.sup.-] be a set of nonpositive integers and [GAMMA] = 2[Z.sup.-] be a set of nonpositive even integers. Then S is a [GAMMA]-semigroup with respect to usual multiplication.

Define a hesitant fuzzy subset H of S such that

[mathematical expression not reproducible] (9)

Clearly, H is a hesitant fuzzy ideal of S.

Example 12. Let S = {a, b, c, d} and [GAMMA] = {[alpha], [beta], [gamma]} be two nonempty sets. Clearly S is a [GAMMA]-semigroup with respect to the operation defined below:
[alpha]   a   b   c   d

a         a   a   a   a
b         a   c   a   c
c         a   b   a   c
d         c   c   c   d

[beta]    a   b   c   d

a         a   a   a   a
b         a   a   a   c
c         a   a   a   c
d         c   c   c   c

[gamma]   a   b   c   d

a         a   a   a   a
b         a   a   a   a
c         a   a   a   a
d         a   a   a   a (10)


Let H be a hesitant fuzzy subset of S such that

H(a) = {0.1, 0.2,0.8,0.9},

H(b) = {0.1, 0.2,0.8},

H(c) = {0.1, 0.2},

H(d) = {0.1}. (11)

Clearly H is a hesitant fuzzy bi-ideal of S. But H is neither a hesitant fuzzy right ideal nor a hesitant fuzzy left ideal of S, since H(b[alpha]d) = H(c) [??] H(b) and H(d[alpha]a) = H(c) [??] H(a), respectively.

Theorem 13. Let H be a nonempty hesitant fuzzy subset of a [GAMMA]-semigroup S. Then the following conditions are equivalent:

(1) H is a hesitant fuzzy bi-ideal of S.

(2) [mathematical expression not reproducible].

Proof. (1) [??] (2) Let H be a hesitant fuzzy bi-ideal of S. Then H is a hesitant fuzzy subsemigroup of S. Let a [member of] S. If some b, c [member of] S and [gamma] [member of] [GAMMA] such that a = b[gamma]c, then we have

[mathematical expression not reproducible] (12)

Thus [mathematical expression not reproducible]. Otherwise, ([H[??].sub.[GAMMA]]YH)(a) = 0 [contains as member or not equal to] H(a). Hence [mathematical expression not reproducible].

Let we prove [mathematical expression not reproducible]. For this, let a [member of] S and let us suppose there exist u, v, p, q [member of] S and [alpha], [beta] [member of] [GAMMA] such that a = u[alpha]v and u = p[beta]q. Since H is a hesitant fuzzy bi-ideal of S, then we have

[mathematical expression not reproducible]. (13)

Thus, [mathematical expression not reproducible].

Otherwise, [mathematical expression not reproducible]. Hence [mathematical expression not reproducible].

(2) [??] (1). Assume that [mathematical expression not reproducible]. Let a, b, c [member of] S and [gamma] [member of] [GAMMA] such that a = b[gamma]c. Then we have

[mathematical expression not reproducible]. (14)

Thus, H is a hesitant fuzzy [GAMMA]-subsemigroup of S. Now let a, b [member of] S and [gamma] [member of] [GAMMA] such that x := a[gamma]b. Since [mathematical expression not reproducible], then it follows that

[mathematical expression not reproducible] (15)

for some [eta], [delta] [member of] [GAMMA]. Hence, H is a hesitant fuzzy bi-ideal of S.

Theorem 14. Let H be a nonempty hesitant fuzzy [GAMMA]-subsemigroup of a [GAMMA]-semigroup S. Then the following conditions are equivalent:

(1) H is a hesitant fuzzy left (resp., right) ideal of S.

(2) [mathematical expression not reproducible] (resp., [mathematical expression not reproducible]).

Proof. (1) [??] (2) Let H be a hesitant fuzzy left ideal of S and a [member of] S. Suppose there exist b, c [member of] S and [gamma] [member of] [GAMMA] such that a = b[gamma]c. Then we have

[mathematical expression not reproducible]. (16)

Since H is a hesitant fuzzy left ideal of S, then it follows that, for all b, c [member of] S and [gamma] [member of] [GAMMA], H(b[gamma]c) [contains as member or equal to] H(c). So in particular, H(c) [subset or equal to] H(a), for all a = b[gamma]c. Thus [mathematical expression not reproducible].

If there do not exist b, c [member of] S and [gamma] [member of] [GAMMA] such that a = b[gamma]c, then (I[??]H)(a) = 0 [subset or equal to] H(a). Hence [mathematical expression not reproducible].

(2) [??] (1). Assume that [mathematical expression not reproducible]. Let a, b [member of] S, [gamma] [member of] [GAMMA], and x := a[gamma]b. Then we have

[mathematical expression not reproducible]. (17)

Hence, H is a hesitant fuzzy left ideal of S. Similarly, we can prove the other case.

Theorem 15. Let S be a regular both-sided [GAMMA]-semigroup, H be a hesitant fuzzy right ideal, and G be a hesitant fuzzy left ideal S. Then [H[??].sub.[GAMMA]]G = H [intersection] G.

Proof. Let H be a hesitant fuzzy right ideal and G be a hesitant fuzzy left ideal of S. Let a [member of] S. Suppose there exist b, c [member of] S and [gamma] [member of] [GAMMA] such that a = b[gamma]c. Then we have

[mathematical expression not reproducible] (18)

If there do not exist b, c [member of] S such that a = b[gamma]c, then ([H[[??].sub.[GAMMA]]YG)(a) = 0 [subset or equal to] (H [intersection] G)(a).

Thus [mathematical expression not reproducible].

For the reverse inclusion, let a [member of] S. Since S is regular, then there exist x [member of] S and [alpha], [beta] [member of] [GAMMA] such that a = a[alpha]x[beta]a := a[gamma]a, where [gamma] := [alpha]x[beta] [member of] [GAMMA]. Therefore,

[mathematical expression not reproducible] (19)

Thus [mathematical expression not reproducible]. Hence [H[??].sub.[GAMMA]G = H [intersection] G.

Let H be a nonempty hesitant fuzzy subset of a [GAMMA]-semigroup S and T [subset or equal to] P([0,1]). Then the set [H.sub.T] := {x [member of] S : H(x) [contains as member or equal to] T} is called the T-cut of H.

Remark 16. Let H be a hesitant fuzzy ideal of a [GAMMA]-semigroup S and [T.sub.1], [T.sub.2] [subset or equal to] P([0,1]) such that [T.sub.1] [subset] [T.sub.2]. Then [mathematical expression not reproducible].

Theorem 17. Let H be a nonempty hesitant fuzzy subset of a [GAMMA]-semigroup S. Then the T-cut [H.sub.T] of H is a left (right) ideal of S for every T [subset or equal to] P([0,1]), provided it is nonempty if and only if H is a hesitant fuzzy left (right) ideal of S.

Proof. For every T [subset or equal to] P([0,1]), let [H.sub.T] be a left ideal of S. We first show that H is a hesitant fuzzy [GAMMA]-subsemigroup of S. If possible there exist [x.sub.0], [y.sub.0] [member of] S, [[gamma].sub.0] [member of] [GAMMA] such that H([x.sub.0][[gamma].sub.0][y.sub.0]) [subset] {H([x.sub.0]) [intersection] H([y.sub.0])}. Let {H([x.sub.0]) [intersection] H([y.sub.0])} = [T.sub.0]. Then H([x.sub.0]) [contains as member or equal to] [T.sub.0], H([y.sub.0]) [contains as member of equal to] To. Thus, [mathematical expression not reproducible], a contradiction. Thus, H([x.sub.0][[gamma].sub.0][y.sub.0]) [contains as member or equal to] {H([x.sub.0]) [intersection] H([y.sub.0])}. Hence H is an hesitant fuzzy [GAMMA]-subsemigroup of S. Again suppose that there exist [x.sub.0], [y.sub.0] [member of] S, [[gamma].sub.0] [member of] [GAMMA] such that H([x.sub.0][[gamma].sub.0][y.sub.0]) [subset] H([y.sub.0]). Since H([y.sub.0]) [subset or equal to] P([0,1]), then let H([y.sub.0]) := [T.sub.0] [subset or equal to] P([0,1]). Thus, [mathematical expression not reproducible], but [mathematical expression not reproducible], a contradiction. Thus, H([x.sub.0][[gamma].sub.0][y.sub.0]) [contains as member or equal to] H([y.sub.0]). Hence H is a hesitant fuzzy left ideal of S.

Conversely, suppose that H is a hesitant fuzzy left ideal of S and T [subset or equal to] P([0,1]) such that [H.sub.T] is nonempty. Let a, b [member of] [H.sub.T] and [gamma] [member of] [GAMMA]. Then H(a) [contains as member or equal to] T and H(b) [contains as member or equal to] T which implies H(a) [intersection] H(b) [contains as member or equal to] T. Since H is a hesitant fuzzy ideal, then it is a hesitant fuzzy [GAMMA]-subsemigroup, and hence H(a[gamma]b) [contains as member or equal to] H(a) n H(b) [intersection] T. Consequently, a[gamma]b [member of] [H.sub.T]. Hence [H.sub.T] is a [GAMMA]-subsemigroup of S. Now, let x [member of] S, y [member of] r and y [member of] HT. Then H(x[gamma]y) [contains as member or equal to] H(y) [contains as member or equal to] T and so x[gamma]y [member of] [H.sub.T]. Hence [H.sub.T] is a left ideal of S. Similarly, we can prove the other case.

Definition 18. A hesitant fuzzy [GAMMA]-subsemigroup H on [GAMMA]-semigroup S is called a hesitant fuzzy interior ideal on S if it satisfies

([for all]x, y, z [member of] S, [alpha], [beta] [member of] [GAMMA])

(H (x[alpha]y[beta]z) [contains as member or equal to] H (y)). (20)

Theorem 19. Let H be a nonempty hesitant fuzzy subset of a [GAMMA]-semigroup S. Then the T-cut [H.sub.T] of H is an interior ideal of S for every T [subset or equal to] P([0,1]), provided it is nonempty if and only if H is a hesitant fuzzy interior ideal of S.

Proof. For every T [subset or equal to] P([0,1]), let [H.sub.T] be an interior ideal of S. Assume that there exist [a.sub.0], [x.sub.0], [y.sub.0] [member of] S, [[beta].sub.0], [[gamma].sub.0] [member of] [GAMMA] such that H([x.sub.0][[beta].sub.0][a.sub.0][[gamma].sub.0][y.sub.0]) [subset] H([a.sub.0]). Since H([a.sub.0]) [subset or equal to] P([0,1]), then let H([a.sub.0]) := [T.sub.0] [subset or equal to] P([0,1]). Thus, [mathematical expression not reproducible], a contradiction. Hence, H([x.sub.0][[beta].sub.0][[beta].sub.0][a.sub.0][[gamma].sub.0][y.sub.0]) [contains as member or equal to] H([a.sub.0]).

Conversely, assume that H is a hesitant fuzzy interior ideal of S and T [subset or equal to] P([0,1]) such that [H.sub.T] is nonempty. Let a, b [member of] S, [beta], [gamma] [member of] [GAMMA] and x [member of] [H.sub.T]. Then H(a[beta]x[gamma]b) [contains as member or equal to] H(x) [contains as member or equal to] T and so a[beta]x[gamma]b [member of] [H.sub.T].

The rest of the proof is a consequence of Theorem 17.

It is well-known that, in a [GAMMA]-semigroup S, every hesitant fuzzy two-sided ideal is a hesitant fuzzy interior ideal of S, but the converse is not true in general. The following example shows that the converse of this property does not hold in general.

Example 20. Let S = [Z.sup.-] be a set of nonpositive integers and [GAMMA] = 2[Z.sup.-] be a set of nonpositive even integers. Then S is a [GAMMA]-semigroup with respect to usual multiplication.

Define a hesitant fuzzy subset H of S such that

[mathematical expression not reproducible] (21)

Clearly, H is a hesitant fuzzy interior ideal of S. But H is not hesitant fuzzy ideal of S, since H((-1)(-2)(-3)) = H(-6) = {0.1} [??] H(-1).

Now we will show that, in a regular [GAMMA]-semigroup, hesitant fuzzy ideals and the hesitant fuzzy interior ideals coincide.

Theorem 21. Let H bea HFS in a regular [GAMMA]-semigroup S. Then H is a hesitant fuzzy ideal of S if and only if H is a hesitant fuzzy interior ideal of S.

Proof. Let H a hesitant fuzzy ideal of S. For any x, y, z [member of] S, and [alpha], [beta] [member of] [GAMMA], we have H(x[alpha]y[beta]z) = H((x[alpha](y[beta]z)) [contains as member or equal to] H(y[beta]z) [contains as member or equal to] H(y). Hence H is a hesitant fuzzy interior ideal of S.

Conversely, let a, b [member of] S and [gamma] [member of] [GAMMA]. Since S is regular, then there exist elements x [member of] S, [alpha], [beta] [member of] [GAMMA] such that a = a[alpha]x[beta]a. Since H is a hesitant fuzzy interior ideal of S, then we have H(a[gamma]b) = H(a[alpha]x[beta]a[gamma]b) [contains as member or equal to] H(a). So H is a hesitant fuzzy right ideal of S. Similarly, we can prove that H is a hesitant fuzzy left ideal of S. Hence H is a hesitant fuzzy ideal of S.

In order to conclude the paper, we obtain the following characterization of a simple [GAMMA]-semigroup by means of a hesitant fuzzy simple [GAMMA]-semigroup.

Definition 22. A [GAMMA]-semigroup S is said to be simple if it does not contain any proper ideal. A [GAMMA]-semigroup S is said to be hesitant fuzzy simple if every hesitant fuzzy ideal of S is a constant function.

Theorem 23. A [GAMMA]-semigroup S is simple if and only if it is hesitant fuzzy simple.

Proof. Suppose that the [GAMMA]-semigroup S is simple. Let H be a hesitant fuzzy ideal of S and x, y [member of] S. Then, by Theorem 17, [H.sub.H(x)] and [H.sub.H(y)] are ideals of S. Since S is simple, then [H.sub.H(x)] = S = [H.sub.H(y)]. Therefore, x, y [member of] [H.sub.H(x)] and x, y [member of] [H.sub.H(y)]. In particular, y [member of] [H.sub.H(x)] and x [member of] [H.sub.H(y)], hence H(y) [contains as member or equal to] H(x) and H(x) [contains as member or equal to] H(y). Thus, H(x) = H(y), for all x, y [member of] S. Hence H is a constant function. Consequently, S is a hesitant fuzzy simple.

Conversely, assume that S is a hesitant fuzzy simple. Let A be any ideal of S. Then its characteristic function [H.sub.A] is a hesitant fuzzy ideal of S and thus [H.sub.A] is a constant function. Let a [member of] S. Since A is nonempty, then [H.sub.A](a) = [0,1] and so a [member of] A. Thus we obtain S = A. Hence S is simple.

Data Availability

The data used to support the findings of this study are included within the article.

https://doi.org/10.1155/2018/5738024

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Mohammad Y. Abbasi (iD), (1) Aakif F. Talee (iD), (1) Sabahat A. Khan (iD), (1) and Kostaq Hila (iD) (2)

(1) Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India

(2) Department of Mathematics & Computer Science, University of Gjirokastra, Gjirokastra 6001, Albania

Correspondence should be addressed to Kostaq Hila; kostaq_hila@yahoo.com

Received 14 May 2018; Revised 25 July 2018; Accepted 14 August 2018; Published 2 September 2018

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Title Annotation:Research Article
Author:Abbasi, Mohammad Y.; Talee, Aakif F.; Khan, Sabahat A.; Hila, Kostaq
Publication:Advances in Fuzzy Systems
Date:Jan 1, 2018
Words:5635
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