# A Hesitant Fuzzy Set Approach to Ideal Theory inΓ-Semigroups

Authors: Mohammad Y. Abbasi; Aakif F. Talee; Sabahat A. Khan; Kostaq Hila
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

We apply the hesitant fuzzy sets theory toΓ-semigroups and provide some characterizations of hesitant fuzzy left (right and bi-) ideals. We introduce the hesitant fuzzy left (resp., right and two-sided) ideal, hesitant fuzzy bi-ideal, and hesitant fuzzy interior ideal in Γ-semigroup and study some properties of them. Finally, a characterization of a simple Γ-semigroup by means of a hesitant fuzzy simple Γ-semigroup is obtained.


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## Body

## 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 Γ-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 Γ-semigroups, by many mathematicians, for instance, Dutta and Adhikari [18, 19], Saha and Sen [20–22], and Hila [23–25]. Mustafa et al. [26] have introduced the Γ- semigroup in terms of intuitionistic fuzzy sets. Sardar, Majumder, Dutta, Davvaz, and Hila [27–30] studied the Γ-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 Γ-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 Γ-semigroup and study some properties of them. Finally, a characterization of a simple Γ-semigroup by means of a hesitant fuzzy simple Γ-semigroup is obtained.


## 2. Preliminaries

In this section, we introduce some necessary notions onΓ-semigroups and present some operations on HFSs that will be used throughout the paper.We first recall the definition of theΓ-semigroup as a generalization of semigroup and ternary semigroup in another way as follows (cf. [17, 22, 24]).Definition 1.
LetM and Γ 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 Γ. Any map from M×Γ×M→M will be called a Γ-multiplication in M and denoted by (·)Γ. The result of this multiplication for a,b∈M and α∈Γ is denoted by aαb. A Γ-semigroup M is an ordered pair (M,(·)Γ) where M and Γ are nonempty sets and (·)Γ is a Γ-multiplication on M which satisfies the following property: ∀(a,b,c,α,β)∈M3×Γ2,(aαb)βc=aα(bβc).This kind of structure is often called one sidedΓ-semigroup or simply Γ-semigroup.Example 2.
LetM be a semigroup and Γ be any nonempty set. Define a mapping M×Γ×M→M by aγb=ab for all a,b∈M and γ∈Γ. Then M is a Γ-semigroup.Example 3.
LetS be the set of all m×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∈P(S), the product AB is not defined. Let Γ be the set of n×m matrices with entries from the same field. Then for A,B,C∈P(S) and P,Q∈Γ, we have APB∈P(S),AQB∈P(S) and since the matrix multiplication is associative, we get that S is a Γ-semigroup.Example 4.
LetM be a set of all negative rational numbers. Obviously M is not a semigroup under usual product of rational numbers. Let Γ={-1/p:pis  prime}. Let a,b,c∈M and α∈Γ. Now if aαb is equal to the usual product of rational numbers a,α,b, then aαb∈M and (aαb)βc=aα(bβc). Hence M is a Γ-semigroup.Example 5.
LetM={-i,0,i} and Γ=M. Then M is a Γ-semigroup under the multiplication over complex numbers while M is not a semigroup under complex number multiplication.These examples show that every semigroup is aΓ-semigroup. Therefore, Γ-semigroups are a generalization of semigroups.A nonempty subsetK of a Γ-semigroup M is called a sub-Γ-semigroup of M if for all a,b∈K and γ∈Γ, aγb∈K.Example 6.
LetM=[0,1] and Γ={1/n∣nis  a  positive  integer}. Then M is a Γ-semigroup under usual multiplication. Let K=[0,1/2]. We have that K is a nonempty subset of M and aγb∈K for all a,b∈K and γ∈Γ. Then K is a sub-Γ-semigroup of M.Definition 7.
The elementa of a Γ-semigroup M is called regular in M if a∈aΓMΓa, where aΓMΓa=aαbβa∣a,b∈M,α,β∈Γ. M is called regular if and only if every element of M is regular.The following is the definition of the so-called both-sidedΓ-semigroup [18].Definition 8.
LetM and Γ 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 Γ. Any map from M×Γ×M→M will be called a Γ-multiplication in M and denoted by (·)Γ. The result of this multiplication for a,b∈M and α∈Γ is denoted by aαb. Any map from Γ×M×Γ→Γ will be called a M-multiplication in Γ and denoted by (·)M. The result of this multiplication for α,β∈Γ and a∈M is denoted by αaβ. A both-sided Γ-semigroup M is an ordered triple (M,(·)Γ,(·)M) where (·)Γ and (·)M satisfy the following property: ∀(a,b,c,α,β,γ)∈M3×Γ3, (1)aαbβc=aαbβc=aαbβcandαaβbγ=αaβbγ=αaβbγ.It is clear that a both-sidedΓ-semigroup is always a one sided Γ-semigroup, while the converse does not hold in general.Clearly, a both-sidedΓ-semigroup S is regular if and only if a∈aΓa, for all a∈S [31].Throughout this paper unless or otherwise mentionedS stands for one sided Γ-semigroup and call it simply Γ-semigroup.LetS 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]: (2)H:S→P0,1where P([0,1]) denotes the set of all subsets of [0,1].LetH and G be any two HFSs on S. We define H⊑G if H(a)⊆G(a), for all a∈S.LetS be a Γ-semigroup. For any HFSs H and G in S, the hesitant fuzzy product of H and G is defined to be the HFS Ho~ΓG on S as follows: (3)Ho~ΓGa=⋃a=bγcHb∩Gcif∃b,c∈S,γ∈Γsuch  thata=bγc∅otherwiseFor any two HFSsH and G on S, the hesitant union H⊔G of H and G is defined to be HFS on S as follows: (4)H⊔G:S→P0,1,a↦Ha∪Gaand the hesitant intersection H⊓G of H and G is defined to be HFS on S as follows: (5)H⊓G:S→P0,1,a↦Ha∩Ga.LetA be any nonempty subset of S. Recall that, we denote by HA the characteristic HFS on S as follows: (6)HA:S→P0,1,a↦0,1ifa∈A∅ifa∉ALetS be a Γ-semigroup. We define a hesitant fuzzy subset I of S as follows: (7)I:S→P0,1∣x→Ix≔0,1.


## 3. Main Results

Definition 9.
LetS be a Γ-semigroup and H be a HFS on S. Then H is called (h1)
a hesitant fuzzyΓ-semigroup on S if it satisfies (∀(x,y,γ)∈S2×Γ)(H(xγy)⊇H(x)∩H(y));(h2)
a hesitant fuzzy left ideal onS if it satisfies ∀x,y,γ∈S2×ΓHxγy⊇Hy;(h3)
a hesitant fuzzy right ideal onS if it satisfies ∀x,y,γ∈S2×ΓHxγy⊇Hx;(h4)
a hesitant fuzzy ideal onS if H is both a hesitant fuzzy left ideal and a hesitant fuzzy right ideal on S;(h5)
a hesitant fuzzy bi-ideal onS if it satisfies (∀(x,y,z,α,β)∈S3×Γ2)(H(xαyβz)⊇H(x)∩H(z)).Example 10.
LetS = M1×2(Z2) be a set of all 1×2 matrices over the field Z2 and Γ = M2×1(Z2) be a set of all 2×1 matrices over the field Z2. Then S is a Γ-semigroup with respect to usual matrix products aαb and αaβ, for all a,b∈S and α,β∈Γ.
Define a hesitant fuzzy subsetH of S such that (8)Hx=0,1ifx=0,00.1otherwise
Clearly,H is a hesitant fuzzy ideal of S.Example 11.
LetS=Z- be a set of nonpositive integers and Γ=2Z- be a set of nonpositive even integers. Then S is a Γ-semigroup with respect to usual multiplication.
Define a hesitant fuzzy subsetH of S such that (9)Hx=0,1ifx=00.1,0.2ifx=-1,-20.1,0.2,0.3otherwise
Clearly,H is a hesitant fuzzy ideal of S.Example 12.
LetS = {a,b,c,d} and Γ={α,β,γ} be two nonempty sets. Clearly S is a Γ-semigroup with respect to the operation defined below:(10)
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aLetH be a hesitant fuzzy subset of S such that (11)Ha=0.1,0.2,0.8,0.9,Hb=0.1,0.2,0.8,Hc=0.1,0.2,Hd=0.1.ClearlyH 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(bαd)=H(c)⊉H(b) and H(dαa)=H(c)⊉H(a), respectively.Theorem 13.
LetH be a nonempty hesitant fuzzy subset of a Γ-semigroup S. Then the following conditions are equivalent:
(1)   H is a hesitant fuzzy bi-ideal of S.
(2)   H ∘ ~ Γ H ⊑ H and H∘~ΓI∘~ΓH⊑H.Proof.
( 1 ) ⇒ ( 2 ) Let H be a hesitant fuzzy bi-ideal of S. Then H is a hesitant fuzzy subsemigroup of S. Let a∈S. If some b,c∈S and γ∈Γ such that a=bγc, then we have (12)H∘~ΓHa=⋃a=bγcHb∩Hc⊆⋃a=bγcHbγc=⋃a=bγcHa=HaThus H∘~ΓH⊑H. Otherwise, (H∘~ΓH)(a)=∅⊆H(a). Hence H∘~ΓH⊑H.
Let we proveH∘~ΓI∘~ΓH⊑H. For this, let a∈S and let us suppose there exist u,v,p,q∈S and α,β∈Γ such that a=uαv and u=pβq. Since H is a hesitant fuzzy bi-ideal of S, then we have (13)H∘~ΓI∘~ΓHa=⋃a=uαvH∘~ΓIu∩Hv=⋃a=uαv⋃u=pβqHp∩Iq∩Hv=⋃a=uαv⋃u=pβqHp∩0,1∩Hv=⋃a=pβqαvHp∩Hv⊆⋃a=pβqαvHpβqαv⊆Ha.Thus, H∘~ΓI∘~ΓH⊑H.
Otherwise, (H∘~ΓI∘~ΓH⊆H)(a)=∅⊆H(a). Hence H∘~ΓI∘~ΓH⊑H.
( 2 ) ⇒ (1). Assume that H∘~ΓH⊑H. Let a,b,c∈S and γ∈Γ such that a=bγc. Then we have (14)Hbγc=Ha⊇H∘~ΓHa=⋃a=bγcHb∩Hc⊇Hb∩Hc.Thus, H is a hesitant fuzzy Γ-subsemigroup of S. Now let a,b∈S and γ∈Γ such that x≔aγb. Since H∘~ΓI∘~ΓH⊑H, then it follows that (15)Haγb≔Hx⊇H∘~ΓI∘~ΓHx=⋃x=mδnH∘~ΓIm∩Hn=⋃x=mδn⋃m=pηqHp∩Iq∩Hn=⋃x=mδn⋃m=pηqHp∩0,1∩Hn=⋃x=pηqδnHp∩Hn⊇Ha∩Hc for some η,δ∈Γ. Hence, H is a hesitant fuzzy bi-ideal of S.Theorem 14.
LetH be a nonempty hesitant fuzzy Γ-subsemigroup of a Γ-semigroup S. Then the following conditions are equivalent:
(1)   H is a hesitant fuzzy left (resp., right) ideal of S.
(2)   I ∘ ~ Γ H ⊑ H (resp., H∘~ΓI⊑H).Proof.
( 1 ) ⇒ ( 2 ) Let H be a hesitant fuzzy left ideal of S and a∈S. Suppose there exist b,c∈S and γ∈Γ such that a=bγc. Then we have (16)I∘~ΓHa=⋃a=bγcIb∩Hc=⋃a=bγc0,1∩Hc=⋃a=bγcHc. Since H is a hesitant fuzzy left ideal of S, then it follows that, for all b,c∈S and γ∈Γ, H(bγc)⊇H(c). So in particular, H(c)⊆H(a), for all a=bγc. Thus (I∘~ΓH)(a)=⋃a=bγcH(c)⊆H(a).
If there do not existb,c∈S and γ∈Γ such that a=bγc, then I∘~Ha=∅⊆Ha. Hence I∘~H⊑H.
( 2 ) ⇒ ( 1 ). Assume that I∘~ΓH⊑H. Let a,b∈S, γ∈Γ, and x≔aγb. Then we have (17)Haγb≔Hx⊇I∘~ΓHx=⋃x=pγqIp∩Hq⊇Ia∩Hb=0,1∩Hb=Hb. Hence, H is a hesitant fuzzy left ideal of S. Similarly, we can prove the other case.Theorem 15.
LetS be a regular both-sided Γ-semigroup, H be a hesitant fuzzy right ideal, and G be a hesitant fuzzy left ideal S. Then H∘~ΓG=H∩G.Proof.
LetH be a hesitant fuzzy right ideal and G be a hesitant fuzzy left ideal of S. Let a∈S. Suppose there exist b,c∈S and γ∈Γ such that a=bγc. Then we have (18)H∘~ΓGa=⋃a=bγcHb∩Gc⊆⋃a=bγcHbγc∩Gbγc=Ha∩Ga=H∩GaIf there do not exist b,c∈S such that a=bγc, then (H∘~ΓG)(a)=∅⊆(H∩G)(a).
ThusH∘~ΓG⊑H∩G.
For the reverse inclusion, leta∈S. Since S is regular, then there exist x∈S and α,β∈Γ such that a=aαxβa≔aγa, where γ≔αxβ∈Γ. Therefore, (19)H∘~ΓGa=⋃a=pδqHp∩Gq⊇Ha∩Ga=H∩Ga
ThusH∘~ΓG⊒H∩G. Hence H∘~ΓG=H∩G.LetH be a nonempty hesitant fuzzy subset of a Γ-semigroup S and T⊆P([0,1]). Then the set HT≔{x∈S:H(x)⊇T} is called the T-cut of H.Remark 16.
LetH be a hesitant fuzzy ideal of a Γ-semigroup S and T1,T2⊆P([0,1]) such that T1⊂T2. Then HT1⊇HT2.Theorem 17.
LetH be a nonempty hesitant fuzzy subset of a Γ-semigroup S. Then the T-cut HT of H is a left (right) ideal of S for every T⊆P([0,1]), provided it is nonempty if and only if H is a hesitant fuzzy left (right) ideal of S.Proof.
For everyT⊆P([0,1]), let HT be a left ideal of S. We first show that H is a hesitant fuzzy Γ-subsemigroup of S. If possible there exist x0,y0∈S,γ0∈Γ such that H(x0γ0y0)⊂{H(x0)∩H(y0)}. Let {H(x0)∩H(y0)}=T0. Then H(x0)⊇T0,H(y0)⊇T0. Thus x0,y0∈HT0 but x0γ0y0∉HT0, a contradiction. Thus, H(x0γ0y0)⊇{H(x0)∩H(y0)}. Hence H is an hesitant fuzzy Γ-subsemigroup of S. Again suppose that there exist x0,y0∈S,γ0∈Γ such that H(x0γ0y0)⊂H(y0). Since H(y0)⊆P([0,1]), then let H(y0)≔T0⊆P([0,1]). Thus, y0∈HT0, but x0γ0y0∉HT0, a contradiction. Thus, H(x0γ0y0)⊇H(y0). Hence H is a hesitant fuzzy left ideal of S.
Conversely, suppose thatH is a hesitant fuzzy left ideal of S and T⊆P([0,1]) such that HT is nonempty. Let a,b∈HT and γ∈Γ. Then H(a)⊇T and H(b)⊇T which implies H(a)∩H(b)⊇T. Since H is a hesitant fuzzy ideal, then it is a hesitant fuzzy Γ-subsemigroup, and hence H(aγb)⊇H(a)∩H(b)⊇T. Consequently, aγb∈HT. Hence HT is a Γ-subsemigroup of S. Now, let x∈S,γ∈Γ and y∈HT. Then H(xγy)⊇H(y)⊇T and so xγy∈HT. Hence HT is a left ideal of S. Similarly, we can prove the other case.Definition 18.
A hesitant fuzzyΓ-subsemigroup H on Γ-semigroup S is called a hesitant fuzzy interior ideal on S if it satisfies (20)∀x,y,z∈S,α,β∈ΓHxαyβz⊇Hy.Theorem 19.
LetH be a nonempty hesitant fuzzy subset of a Γ-semigroup S. Then the T-cut HT of H is an interior ideal of S for every T⊆P([0,1]), provided it is nonempty if and only if H is a hesitant fuzzy interior ideal of S.Proof.
For everyT⊆P([0,1]), let HT be an interior ideal of S. Assume that there exist a0,x0,y0∈S,β0,γ0∈Γ such that H(x0β0a0γ0y0)⊂H(a0). Since H(a0)⊆P([0,1]), then let H(a0)≔T0⊆P([0,1]). Thus, a0∈HT0, but x0β0a0γ0y0∉HT0, a contradiction. Hence, H(x0β0a0γ0y0)⊇H(a0).
Conversely, assume thatH is a hesitant fuzzy interior ideal of S and T⊆P([0,1]) such that HT is nonempty. Let a,b∈S,β,γ∈Γ and x∈HT. Then H(aβxγb)⊇H(x)⊇T and so aβxγb∈HT.
The rest of the proof is a consequence of Theorem17.It is well-known that, in aΓ-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.
LetS=Z- be a set of nonpositive integers and Γ=2Z- be a set of nonpositive even integers. Then S is a Γ-semigroup with respect to usual multiplication.
Define a hesitant fuzzy subsetH of S such that (21)Hx=0,1ifx=00.1,0.2ifx=-1,-20.1ifx=-3,-60.1,0.2,0.3otherwiseClearly,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 regularΓ-semigroup, hesitant fuzzy ideals and the hesitant fuzzy interior ideals coincide.Theorem 21.
LetH be a HFS in a regular Γ-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.
LetH a hesitant fuzzy ideal of S. For any x, y, z∈S, and α,β∈Γ, we have H(xαyβz)=H((xα(yβz))⊇H(yβz)⊇H(y). Hence H is a hesitant fuzzy interior ideal of S.
Conversely, leta,b∈S and γ∈Γ. Since S is regular, then there exist elements x∈S,α,β∈Γ such that a = aαxβa. Since H is a hesitant fuzzy interior ideal of S, then we have H(aγb)=H(aαxβaγb)⊇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 simpleΓ-semigroup by means of a hesitant fuzzy simple Γ-semigroup.Definition 22.
AΓ-semigroup S is said to be simple if it does not contain any proper ideal. A Γ-semigroup S is said to be hesitant fuzzy simple if every hesitant fuzzy ideal of S is a constant function.Theorem 23.
AΓ-semigroup S is simple if and only if it is hesitant fuzzy simple.Proof.
Suppose that theΓ-semigroup S is simple. Let H be a hesitant fuzzy ideal of S and x,y∈S. Then, by Theorem 17, HH(x) and HH(y) are ideals of S. Since S is simple, then HH(x)=S=HH(y). Therefore, x,y∈HH(x) and x,y∈HH(y). In particular, y∈HH(x) and x∈HH(y), hence H(y)⊇H(x) and H(x)⊇H(y). Thus, H(x)=H(y), for all x,y∈S. Hence H is a constant function. Consequently, S is a hesitant fuzzy simple.
Conversely, assume thatS is a hesitant fuzzy simple. Let A be any ideal of S. Then its characteristic function HA is a hesitant fuzzy ideal of S and thus HA is a constant function. Let a∈S. Since A is nonempty, then HA(a)=[0,1] and so a∈A. Thus we obtain S=A. Hence S is simple.



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