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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 169-175

Published online June 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.2.169

© The Korean Institute of Intelligent Systems

## A New Generalization of Hesitant and Interval-Valued Fuzzy Ideals of Ternary Semigroups

Pongpun Julatha1 and Aiyared Iampan2

1Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand
2Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, Mae Ka, University of Phayao, Phayao, Thailand

Correspondence to :
Aiyared Iampan (aiyared.ia@up.ac.th)

Received: March 14, 2021; Revised: June 1, 2021; Accepted: June 12, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The main aim of this article is to introduce the concept of a sup-hesitant fuzzy ideal, which is a generalization of a hesitant fuzzy ideal and an interval-valued fuzzy ideal, in a ternary semigroup. Some characterizations of a sup-hesitant fuzzy ideal are examined in terms of a fuzzy set, a hesitant fuzzy set, and an interval valued fuzzy set. Further, we discuss the relation between an ideal and a generalization of a characteristic hesitant fuzzy set and a characteristic interval-valued fuzzy set.

Keywords: Ternary semigroup, sup-hesitant fuzzy ideal, Hesitant fuzzy ideal, Interval-valued fuzzy ideal

### 1. Introduction

Ternary algebraic structures were introduced by Lehmer [1] in 1932, who examined exact ternary algebraic structures called triplexes, which turned out to be ternary groups. Ternary semigroups were first introduced by Stefan Banach, who showed that a ternary semigroup does not necessarily reduce to a semigroup. In 1965, Sioson [2] studied ideal theory in ternary semigroups. In addition, Iampan [3] studied the lateral ideal of a ternary semigroup in 2007. Ideal theory is an important concept for studying ternary semigroups and algebraic structures.

After the concept of a fuzzy set was introduced by Zadeh [4], the ideal theory in a ternary semigroup was extended to fuzzy ideal theory, bipolar fuzzy ideal theory, interval-valued fuzzy ideal theory, and hesitant fuzzy ideal theory in a ternary semigroup. In 2012, Kar and Sarkar [5] introduced a fuzzy left (lateral, right) ideal and fuzzy ideal of a ternary semigroup and used a fuzzy set to characterize a regular (intra-regular) ternary semigroup. In 2015, Ansari and Masmali [6] studied the bipolar (λ, δ)-fuzzy ideal of a ternary semigroup. In 2016, Jun et al. [7] introduced a hesitant fuzzy semigroup with a frontier and studied the hesitant union and hesitant intersection of two hesitant fuzzy semigroups with a frontier. Muhiuddin [8] introduced a hesitant fuzzy G-filter for a residuated lattice and provided some conditions for a hesitant fuzzy filter to be a hesitant fuzzy G-filter. In 2018, Suebsung and Chinram [9] studied an interval-valued fuzzy ideal extension of a ternary semigroup. In 2019, Muhiuddin et al. [10] introduced an (α?, β?)-fuzzy left (right, lateral) ideal in a ternary semigroup. In addition, in 2020, Talee et al. [11] introduced a hesitant fuzzy ideal and a hesitant fuzzy interior ideal in an ordered Γ-semigroup and characterized a simple ordered Γ-semigroup in terms of a hesitant fuzzy simple ordered Γ-semigroup.

The main aim of this article is to introduce the concept of a sup-hesitant fuzzy ideal of a ternary semigroup, which is a generalization of a hesitant fuzzy ideal and an interval-valued fuzzy ideal in a ternary semigroup. Some characterizations of an sup-hesitant fuzzy ideal are examined in terms of a fuzzy set, a hesitant fuzzy set, and an interval valued fuzzy set. Further, we discuss the relation between an ideal and a generalization of a characteristic hesitant fuzzy set and a characteristic interval-valued fuzzy set.

### 2. Preliminaries

In the following sections, we introduce some definitions and results that are important for the present study.

By a ternary semigroup, we mean a set T ≠ ∅ with a ternary operation T ×T ×TT, written as (t1, t2, t3) ? t1t2t3 satisfying the identity (for all t1, t2, t3, t4, t5T)((t1t2t3)t4t5 = t1(t2t3t4)t5 = t1t2(t3t4t5)). Throughout this paper, T is represented as a ternary semigroup. Let X ≠ ∅, Y ≠ ∅, and Z ≠ ∅ be subsets of T. We define the subset XYZ of T as follows: XYZ = {xyz | xX, yY, zZ}. A subset A ≠ ∅ of T is said to be a left (lateral, right) ideal (L(Lt, R)I) of T and TTAA (TATA, ATTA). If the subset is an LI, LtI, and RI of T, then it is said to be an ideal (Id) of T.

A fuzzy set (FS) f [4] in set X ≠ ∅ is a mapping from X to the unit segment of the real line [0, 1]. Kar and Sarkar [5] studied an FS in a ternary semigroup and introduced the concepts of a fuzzy left (lateral, right) ideal and a fuzzy ideal of ternary semigroups as follows:

### Definition 2.1 [5]

Let f be the FS in T. Then, f is said to be

• (1) a fuzzy left ideal (FLI) of T while (for all t1, t2, t3T)(f(t3) ≤ f(t1t2t3)),

• (2) a fuzzy lateral ideal (FLtI) of T while (for all t1, t2, t3T)(f(t2) ≤ f(t1t2t3)),

• (3) a fuzzy right ideal (FRI) of T while (for all t1, t2, t3T)(f(t1) ≤ f(t1t2t3)), or

• (4) a fuzzy ideal (FI) of T while f is an FLI, an FLtI, and an FRI of T, that is, (for all t1, t2, t3T)(max{f(t1), f(t2), f(t3)} ≤ f(t1t2t3)).

Let [[0, 1]] be the set of all closed subintervals of [0, 1]; that is

$[[0,1]]={[t-,t+]?t-,t+∈[0,1] and t-≤t+}.$

Let $t1^=[t1-,t1+],t2^=[t2-,t2+]∈[[0,1]]$. We define the operations ?, =, ?, and rmax as follows:

• (1) $t1^?t2^⇔t1-≤t2-,t1+≤t2+$,

• (2) $t1^=t2^⇔t1-=t2-,t1+=t2+$,

• (3) $t1^?t2^⇔t1^?t2^,t1^≠t2^$,

• (4) $rmax{t1^,t2^}=[max{t1-,t2-},max{t1+,t2+}]$.

Let X ≠ ∅ be a set. A mapping ν?: X → [[0, 1]] is said to be an interval-valued fuzzy set (IvFS) [12] on X, where for any xX, ν?(x) = [ν(x), ν+(x)], anything ν and ν+ are FSs in X such that ν (x) ≤ ν+(x).

For a subset A of X, the characteristic interval-valued fuzzy set CIA of A on X is defined by

$CIA:X→[[0,1]],x?{1^if x∈A,0^otherwise,$

where 0? = [0, 0] and 1? = [1, 1].

### Definition 2.2 [9]

Let ν? be an IvFS on T. Then, ν? is said to be

• (1) an interval-valued fuzzy left ideal (IvFLI) of T while (for all t1, t2, t3T)(ν?(t3) ? ν?(t1t2t3)),

• (2) an interval-valued fuzzy lateral ideal (IvFLtI) of T while (for all t1, t2, t3T)(ν?(t2) ? ν?(t1t2t3)),

• (3) an interval-valued fuzzy right ideal (IvFRI) of T while (for all t1, t2, t3T)(ν?(t1) ? ν?(t1t2t3)),

• (4) an interval-valued fuzzy ideal (IvFI) of T while it is an IvFLI, an IvFLtI, and an IvFRI of T, that is, (for all t1, t2, t3T)(rmax{ν?(t1), ν?(t2), ν?(t3)} ? ν?(t1t2t3)).

### Theorem 2.3 [9]

A subset A ≠ ∅ of T is an Id of T if and only if CIA is an IvFI of T.

Torra and his colleague [13,14] defined a hesitant fuzzy set (HFS) on a set X ≠ ∅ in terms of a mapping h that, when applied to X, returns a subset of [0, 1], that is, h: X → ℘[0, 1], where ℘[0, 1] denotes the set of all subsets of [0, 1]. Talee et al. [15] applied the concept of an HFS to a ternary semigroup and introduced the concepts of a hesitant fuzzy left (lateral, right) ideal and a hesitant fuzzy ideal of a ternary semigroup as follows:

### Definition 2.4 [15]

Let h be an HFS on T. Then, h is said to be

• (1) a hesitant fuzzy left ideal (HFLI) of T while (for all t1, t2, t3T)(h(t3) ⊆ h(t1t2t3)),

• (2) a hesitant fuzzy lateral ideal (HFLtI) of T while (for all t1, t2, t3T)(h(t2) ⊆ h(t1t2t3)),

• (3) a hesitant fuzzy right ideal (HFRI) of T while (for all t1, t2, t3T)(h(t1) ⊆ h(t1t2t3)),

• (4) a hesitant fuzzy ideal (HFI) of T while it is a HFLI, a HFLtI, and a HFRI of T, that is, (for all t1, t2, t3T)(h(t1) ∪ h(t2) ∪ h(t3) ⊆ h(t1t2t3)).

For a subset A of a set X ≠ ∅, define the characteristic hesitant fuzzy set (CHFS) CHA of A on X as follows:

$CHA:X→P[0,1],x?{[0,1]while x∈A,∅otherwise.$

### Theorem 2.5 [15]

A subset A ≠ ∅ of T is an Id of T if and only if CHA is an HFI of T.

It is well known that an HFS on T is a generalization of the concept of an IvFS on T. In general, we can see that the HFI of T is not an IvFI of T, and an IvFI of T is not an HFI of T, as shown in Example 2.6.

### Example 2.6

Consider a ternary semigroup T = {−i, 0, i} under the usual multiplication over a complex number.

• (1) Define an HFS h on T by h(i) = h(−i) = {0.1, 0.2, 0.3, 0.5} and h(0) = [0.1, 0.5], and we have h as an HFI of T but not an IvFI of T because h is not an IvFS on T.

• (2) Define an IvFS ν? on T by ν?(−i) = ν?(i) = [0, 0.5] and ν?(0) = [0.5, 1], and we have ν? as an IvFI of T but not an HFI of T because

$ν^(i)∪ν^(-i)∪ν^(0)=[0,1]⊄[0.5,1]=ν^(0)=ν^((i)(0)(-i)).$

• (3) Define an IvFS g on T by g(i) = g(−i) = [0, 0.4] and g(0) = [0, 1]. Then, g is both an HFI and an IvFI of T.

### 3. Main Results

For ∇ ∈ ℘[0, 1], define SUP ∇ by

$SUP∇={sup ∇while ∇≠∅,0otherwise.$

For an HFS h on X and ∇ ∈ ℘[0, 1], we define SUP [h; ∇] as

$SUP [h;∇]={x∈X?SUP h(x)≥SUP ∇}.$

### Definition 3.1

Given ∇ ∈ ℘[0, 1], an HFS h on T is said to be a sup-hesitant fuzzy left (lateral, right) ideal of T related to ∇ (∇-sup-HFL(Lt, R)I of T), whereas the set SUP [h; ∇] is an L(Lt, R)I of T. If h is a ∇-sup-HFL(Lt, R)I of T for all ∇ ∈ ℘[0, 1] when SUP [h; ∇] ≠ ∅, then h is said to be a sup-hesitant fuzzy left (lateral, right) ideal (sup-HFL(Lt, R)I) of T.

### Definition 3.2

An HFS h on T is said to be a sup-hesitant fuzzy ideal of T related to ∇ (∇-sup-HFI of T), whereas it is an ∇-sup-HFLI, a ∇-sup-HFLtI, and a ∇-sup-HFRI of T. If h is a ∇-sup-HFI of T for all ∇ ∈ ℘[0, 1] when SUP [h; ∇] ≠ ∅, then h is said to be a sup-hesitant fuzzy ideal (sup-HFI) of T.

### Lemma 3.3

All IvFL(Lt, R)Is of T are a sup-HFL(Lt, R)I.

Proof

Suppose that ν? is an IvFLI of T and ∇ ∈ ℘[0, 1] such that SUP [ν?; ∇] ≠ ∅. Let a, bT, and let c ∈ SUP [ν?; ∇]. Then, sup ν? (c) ≥ SUP ∇. Because ν? is an IvFLI of T, we have

$SUP ∇≤sup ν^(c)=ν+(c)≤ν+(abc)=sup ν^(abc).$

Thus, abc ∈ SUP [ν?; ∇]. Hence, SUP [ν?; ∇] is an LI of T, which indicates that ν? is a ∇-sup-HFLI of T. Therefore, we conclude that ν? is a sup-HFLI of T.

From Lemma 3.3, we obtain Theorem 3.4.

### Theorem 3.4

All IvFIs of T are a sup-HFI.

The converses of Lemma 3.3 and Theorem 3.4 are not true, as shown in Example 3.5.

### Example 3.5

Consider a ternary semigroup T = {O, A, B, C, D, I} under the usual matrix multiplication, where

$O=(0000),A=(1000),B=(0100),C=(0010),D=(0001),I=(1001).$

Define an IvFS ν? on T by

$ν^(O)=[0,1], ν^(A)=[0.4,1], ν^(B)=[0.6,1],ν^(C)=ν^(D)=[0.5,1], ν^(I)=0^.$

Thus, ν? is a sup-HFI of T but not an IvFI of T. Moreover, we know that

• (1) ν? is not an IvFLI of T because ν?(OAB) = [0, 1] ? [0.6, 1] = ν?(B).

• (2) ν? is not an IvFLtI of T because ν?(OAB) = [0, 1] ? [0.4, 1] = ν?(A).

• (3) ν? is not an IvFRI of T because ν?(CBO) = [0, 1] ? [0.5, 1] = ν?(C).

From Lemma 3.3, Theorem 3.4, and Example 3.5, we find that in an arbitrary ternary semigroup, a sup-HFL(Lt, R)I is a generalization of the concept of an IvFL(Lt, R)I, and a sup-HFI is a generalization of the concept of an IvFI.

### Lemma 3.6

All HFL(Lt, R)Is of T are a sup-HFL(Lt, R)I.

Proof

Suppose that h is an HFLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. Then, SUP h(c) ≥ SUP ∇. Because h is an HFLI of T, we have h(c) ⊆ h(abc) and thus SUP h(c) ≤ SUP h(abc). Therefore, abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T, which signifies that h is a ∇-sup-HFLI of T. We thus conclude that h is a sup-HFLI of T.

From Lemma 3.6, we obtain Theorem 3.7.

### Theorem 3.7

All HFIs of T are a sup-HFI.

Example 3.8 shows that the converses of Lemmas 3.6 and Theorem 3.7 do not hold.

### Example 3.8

Consider a ternary semigroup T = {O, I, X, Y, Z} under the usual matrix multiplication, where

$O=(000000000), I=(100010001), X=(000010000),Y=(001000000),Z=(000000100).$

Define an HFS h on T as

$h(O)={0,1}, h(X)=[0,1], h(Y)=h(Z)=[0,1),h(I)=∅.$

Thus, h is a sup-HFI of T, but not an HFI of T. Moreover, we know that

• (1) h is not an HFLI of T because h(X) = [0, 1] ⊃ {0, 1} = h(OYX).

• (2) h is not an HFLtI of T because h(X) = [0, 1] ⊃ {0, 1} = h(OXI).

• (3) h is not an HFRI of T because h(X) = [0, 1] ⊃ {0, 1} = h(XOZ).

From Lemma 3.6, Theorem 3.7, and Example 3.8, we find that in an arbitrary ternary semigroup, a sup-HFL(Lt, R)I is a generalization of the concept of an HFL(Lt, R)I, and a sup-HFI is a generalization of the concept of an HFI.

Let h be an HFS on T, and define the FS Fh in T as

$Fh:T→[0,1],x?SUP h(x).$

The following lemma characterizes the sup-types of HFSs on T by FS Fh.

### Lemma 3.9

An HFS h on T is a sup-HFL(Lt, R)I of T if and only if Fh is an FL(Lt, R)I of T.

Proof

Suppose that h is an sup-HFLI of T. Let a, b, cT, and let ∇ = h(c). Then, c ∈ SUP [h; ∇]. Thus, h is a ∇-sup-HFLI of T, which indicates that SUP [h; ∇] is an LI of T. Hence, abc ∈ SUP [h; ∇] and thus

$Fh(abc)=SUP h(abc)≥SUP ∇=SUP h(c)=Fh(c).$

Therefore, Fh is an FLI of T.

Conversely, suppose that Fh is an FLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. Then,

$SUP h(abc)=Fh(abc)≥Fh(c)=SUP h(c)≥SUP ∇,$

and it is implied that abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T; that is, h is a ∇-sup-HFLI of T. Therefore, we conclude that h is a sup-HFLI of T.

Let h be an HFS on T and ∇ ∈ ℘[0, 1], and we define the HFS H(h; ∇) on T as

$(for all x∈T)(H(h;∇)(x)={t∈∇?SUP h(x)≥t}).$

We then denote H(h;?xT h(x)) by Hh and H(h; [0, 1]) by Ih. Then, Ih is an IvFS on T.

### Remark 3.10

If h is an HFS on T, then h(x) ⊆ Hh(x) ⊆ Ih(x) and SUP h(x) = SUPHh(x) = supIh(x) for all xT.

Now, we study sup-types of HFSs on T using the HFS H(h; ∇) and the IvFS Ih.

### Lemma 3.11

An HFS h on T is a sup-HFL(Lt, R)I of T if and only if H(h; ∇) is a HFL(Lt, R)I of T for all ∇ ∈ ℘[0, 1].

Proof

Suppose that h is a sup-HFLI of T and ∇ ∈ ℘[0, 1]. Let a, b, cT. If H(h; ∇)(c) is empty, then H(h; ∇)(c) ⊆ H(h; ∇)(abc). In addition, let t ∈ H(h; ∇)(c). Then, t ∈ ∇, SUP h(c) ≥ t, and c ∈ SUP [h; h(c)]. Because h is a sup-HFLI of T, we have SUP [h; h(c)] as an LI of T. Hence, abc ∈ SUP [h; h(c)], which indicates that SUP h(abc) ≥ SUP h(c) ≥ t. Thus, t ∈ H(h; ∇)(abc). Therefore, H(h; ∇)(c) ⊆ H(h; ∇)(abc). Consequently, H(h; ∇) is an HFLI of T.

Conversely, suppose that H(h; ∇) is an HFLI of T for all ∇ ∈ ℘[0, 1]. Let a, b, cT and ∇ ∈ ℘[0, 1] exist such that c ∈ SUP [h; ∇]. Then, H(h; ∇)(c) = ∇, and by assumption, we have ∇ = H(h; ∇)(c) ⊆ H(h; ∇)(abc). Thus, SUP h(abc) ≥ SUP ∇, and it is implied that abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T; that is, h is a ∇-sup-HFLI of T. Therefore, we conclude that h is a sup-HFLI of T.

### Theorem 3.12

For an HFS h on T, the following statements are equivalent.

• (1) h is a sup-HFL(Lt, R)I of T.

• (2) Hh is an HFL(Lt, R)I of T.

• (3) Hh is a sup-HFL(Lt, R)I of T.

• (4) Ih is an IvFL(Lt, R)I of T.

• (5) Ih is a sup-HFL(Lt, R)I of T.

• (6) Ih is an HFL(Lt, R)I of T.

Proof

(1) ⇒ (2) and (1) ⇒ (6). These follow from Lemma 3.11.

(2) ⇒ (3) and (6) ⇒ (5). These follow from Lemma 3.6.

(4) ⇒ (5). This follows from Lemma 3.3.

(3) ⇒ (1). Suppose that Hh is an sup-HFLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. Based on Remark 3.10, we have SUPHh(c) = SUP h(c) ≥ SUP ∇ and thus c ∈ SUP [Hh; ∇]. We assume that SUP [Hh; ∇] is an LI of T, and then abc ∈ SUP [Hh; ∇]. By Remark 3.10 again, we can see that SUP h(abc) = SUPHh (abc) ≥ SUP ∇, which signifies that abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T; that is, h is a ∇-sup-HFLI of T. We therefore conclude that h is a sup-HFLI of T.

(1) ⇒ (4). Suppose that h is a sup-HFLI of T and a, b, cT. Then, c ∈ SUP [h; h(c)], and therefore by assumption we have abc ∈ SUP [h; h(c)]. Thus, SUP h(c) ≤ SUP h(abc), and therefore Ih(c) = [0, SUP h(c)] ? [0, SUP h(abc)] = Ih(abc). Hence, Ih is an IvFLI of T.

(5) ⇒ (1). Let Ih be an sup-HFLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. By Remark 3.10, we have sup Ih(c) = SUPh(c) ≥ SUP ∇, and thus c ∈ SUP [Ih; ∇]. We assume that abc ∈ SUP [Ih; ∇]. By Remark 3.10, we obtain SUP h(abc) = supIh (abc) ≥ SUP ∇, which indicates that abc ∈ SUP [h; ∇]. Hence, SUP[h; ∇] is an LI of T, which signifies that h is a ∇-sup-HFLI of T. Therefore, we conclude that h is a sup-HFLI of T.

From Lemma 3.9 and Theorem 3.12, we obtain Theorem 3.13.

### Theorem 3.13

For an HFS h on T, the following statements are equivalent.

• (1) h is a sup-HFI of T.

• (2) (for all a, b, cT)(SUP h(abc) ≥ max{SUP h(a), SUP h(b), SUP h(c)}).

• (3) Fh is an FI of T.

• (4) Hh is an HFI of T.

• (5) Hh is a sup-HFI of T.

• (6) Ih is an IvFI of T.

• (7) Ih is a sup-HFI of T.

• (8) Ih is an HFI of T.

For a subset A of T and ∇, Ω ∈ ℘[0, 1] with SUP ∇ < SUP Ω, we define a map $HA(∇,Ω)$ as follows:

$HA(∇,Ω):T→P[0,1],x?{Ωwhile x∈A,∇otherwise.$

Then, $HA(∇,Ω)$ is an HFS on T, which is said to be a sup (∇, Ω)-characteristic hesitant fuzzy set (sup (∇, Ω)-CHFS) of A of T. In addition, sup (∇, Ω)-CHFS with ∇ = ∅ and Ω = [0, 1] is the CHFS of A, that is, $HA(∅,[0,1])=CHA$. Moreover, sup (∇, Ω)-CHFS with ∇ = 0? and Ω = 1? is the CIvFS of A, that is, $HA(0^,1^)=CIA$.

### Theorem 3.14

Let a subset A ≠ ∅ of T and ∇, Ω ∈ ℘[0, 1] exist such that SUP ∇ < SUP Ω. Then, A is an Id of T if and only if $HA(∇,Ω)$ is an sup-HFI of T.

Proof

Suppose that there exist a, b, cT such that

$SUP HA(∇,Ω)(abc). Then,

$HA(∇,Ω)(a)=Ω, HA(∇,Ω)(b)=Ω$, or $HA(∇,Ω)(c)=Ω$, which signifies that aA, bA, or cA. Because A is an Id of T, we have abcA and $HA(∇,Ω)(abc)=Ω$. Thus,

$SUP HA(∇,Ω)(abc)=max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(b),SUP HA(∇,Ω)(c)}$

is a contradiction. Hence,

$SUP HA(∇,Ω)(abc)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(b),SUP HA(∇,Ω)(c)}$

for all a, b, cT, and by Theorem 3.13, we have $HA(∇,Ω)$ being a sup-HFI of T.

Conversely, let aA and x, yT. Then $HA(∇,Ω)(a)=Ω$. Because $HA(∇,Ω)$ is a sup-HFI of T, and by Theorem 3.13, we have

$SUP HA(∇,Ω)(axy)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(x),SUP HA(∇,Ω)(y)},SUP HA(∇,Ω)(xay)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(x),SUP HA(∇,Ω)(y)},$

and

$SUP HA(∇,Ω)(xya)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(x),SUP HA(∇,Ω)(y)}=SUP Ω.$

Thus,

$SUP HA(∇,Ω)(axy)=SUP HA(∇,Ω)(xay)=SUP HA(∇,Ω)(xya)=SUP Ω,$

which indicates that axy, xay, xyaA. Hence, A is the Id of T.

From Theorems 2.3, 2.5, 3.4, 3.7, and 3.14, we obtain Theorem 3.15.

### Theorem 3.15

For a subset A ≠ ∅ of T, the following statements are equivalent.

• (1) A is an Id of T.

• (2) CIA is an IvFI of T.

• (3) CIA is a sup-HFI of T.

• (4) CHA is an HFI of T.

• (5) CHA is a sup-HFI of T.

• (6) $HA(∇,Ω)$ is a sup-HFI of T for all ∇, Ω ∈ P[0, 1] with SUP ∇ < SUP Ω

### 4. Conclusion

In this paper, we introduced the concept of a sup-HFI in a ternary semigroup, which is a generalization of an HFI and an IvFI in a ternary semigroup, and examined some characterizations of a sup-HFI in terms of an FS, an HFS, and an IvFS. Further, we discussed the relation between an Id and the generalizations of CHFSs and CIvFSs. As important study results, we found that the following statements are all equivalent in a ternary semigroup T: A subset A is an Id, CIA is an IvFI, CIA is a sup-HFI, CHA is an HFI, and CHA is a sup-HFI.

In the future, we will study a sup-HFI in a Γ semigroup and examine some characterizations of a sup-HFI in terms of an FS, an HFS, and an IvFS.

### Conflict of Interest

No potential conflict of interest relevant to this article was reported.

### References

1. Lehmer, DH (1932). A ternary analogue of abelian groups. American Journal of Mathematics. 54, 329-338. https://doi.org/10.2307/2370997
2. Sioson, FM (1965). Ideal theory in ternary semigroups. Math Japon. 10, 63-84.
3. Iampan, A (2007). Lateral ideals of ternary semigroups. Ukrainian Mathematical Bulletin. 4, 525-534.
4. Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
5. Kar, S, and Sarkar, P (2012). Fuzzy ideals of ternary semigroups. Fuzzy Information and Engineering. 4, 181-193. https://doi.org/10.1007/s12543-012-0110-4
6. Ansari, MA, and Masmali, IA (2015). Ternary Semigroups in Terms of Bipolar (λ, δ)-Fuzzy Ideals. International Journal of Algebra. 9, 475-486. http://doi.org/10.12988/ija.2015.5740
7. Jun, YB, Song, SZ, and Muhiuddin, G (2016). Hesitant fuzzy semigroups with a frontier. Journal of Intelligent & Fuzzy Systems. 30, 1613-1618. http://doi.org/10.3233/IFS-151869
8. Muhiuddin, G (2016). Hesitant fuzzy filters and hesitant fuzzy G-filters in residuated lattices. Journal of Computational Analysis & Applications. 21, 394-404.
9. Suebsung, S, and Chinram, R (2018). Interval valued fuzzy ideal extensions of ternary semigroups. International Journal of Mathematics and Computer Science. 13, 15-27.
10. Muhiuddin, G, Rehman, N, and Jun, YB (2019). A generalization of (∈, ∈ ∨ q)-fuzzy ideals in ternary semigroups. Annals Communications in Mathematics. 2, 73-83.
11. Talee, AF, Abbasi, MY, Muhiuddin, G, and Khan, SA (2020). Hesitant fuzzy sets approach to ideal theory in ordered Γ-semigroups. Italian Journal of Pure and Applied Mathematics. 43, 73-85.
12. Zadeh, LA (1975). The concept of a linguistic variable and its application to approximate reasoning?I. Information Sciences. 8, 199-249. https://doi.org/10.1016/0020-0255(75)90036-5
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### Biographies

Pongpun Julatha is a faculty member of the Faculty of Science and Technology, Pibulsongkram Rajabhat University, Thailand. He received his B.S. and M.S. degrees and his Ph.D. in mathematics from Naresuan University, Thailand. His areas of interest include the algebraic theory of semigroups, ternary semigroups, and Γ-semigroups and fuzzy algebraic structures.

E-mail: pongpun.j@psru.ac.th

Aiyared Iampan is an associate professor at the Department of Mathematics, University of Phayao, Thailand. He received his B.S. and M.S. degrees and his Ph.D. in mathematics from Naresuan University, Thailand. His areas of interest include algebraic theory of semigroups, ternary semigroups, and Γ-semigroups, fuzzy algebraic structures, and logical algebras.

E-mail: aiyared.ia@up.ac.th

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 169-175

Published online June 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.2.169

Copyright © The Korean Institute of Intelligent Systems.

## A New Generalization of Hesitant and Interval-Valued Fuzzy Ideals of Ternary Semigroups

Pongpun Julatha1 and Aiyared Iampan2

1Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand
2Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, Mae Ka, University of Phayao, Phayao, Thailand

Correspondence to:Aiyared Iampan (aiyared.ia@up.ac.th)

Received: March 14, 2021; Revised: June 1, 2021; Accepted: June 12, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

The main aim of this article is to introduce the concept of a sup-hesitant fuzzy ideal, which is a generalization of a hesitant fuzzy ideal and an interval-valued fuzzy ideal, in a ternary semigroup. Some characterizations of a sup-hesitant fuzzy ideal are examined in terms of a fuzzy set, a hesitant fuzzy set, and an interval valued fuzzy set. Further, we discuss the relation between an ideal and a generalization of a characteristic hesitant fuzzy set and a characteristic interval-valued fuzzy set.

Keywords: Ternary semigroup, sup-hesitant fuzzy ideal, Hesitant fuzzy ideal, Interval-valued fuzzy ideal

### 1. Introduction

Ternary algebraic structures were introduced by Lehmer [1] in 1932, who examined exact ternary algebraic structures called triplexes, which turned out to be ternary groups. Ternary semigroups were first introduced by Stefan Banach, who showed that a ternary semigroup does not necessarily reduce to a semigroup. In 1965, Sioson [2] studied ideal theory in ternary semigroups. In addition, Iampan [3] studied the lateral ideal of a ternary semigroup in 2007. Ideal theory is an important concept for studying ternary semigroups and algebraic structures.

After the concept of a fuzzy set was introduced by Zadeh [4], the ideal theory in a ternary semigroup was extended to fuzzy ideal theory, bipolar fuzzy ideal theory, interval-valued fuzzy ideal theory, and hesitant fuzzy ideal theory in a ternary semigroup. In 2012, Kar and Sarkar [5] introduced a fuzzy left (lateral, right) ideal and fuzzy ideal of a ternary semigroup and used a fuzzy set to characterize a regular (intra-regular) ternary semigroup. In 2015, Ansari and Masmali [6] studied the bipolar (λ, δ)-fuzzy ideal of a ternary semigroup. In 2016, Jun et al. [7] introduced a hesitant fuzzy semigroup with a frontier and studied the hesitant union and hesitant intersection of two hesitant fuzzy semigroups with a frontier. Muhiuddin [8] introduced a hesitant fuzzy G-filter for a residuated lattice and provided some conditions for a hesitant fuzzy filter to be a hesitant fuzzy G-filter. In 2018, Suebsung and Chinram [9] studied an interval-valued fuzzy ideal extension of a ternary semigroup. In 2019, Muhiuddin et al. [10] introduced an (α?, β?)-fuzzy left (right, lateral) ideal in a ternary semigroup. In addition, in 2020, Talee et al. [11] introduced a hesitant fuzzy ideal and a hesitant fuzzy interior ideal in an ordered Γ-semigroup and characterized a simple ordered Γ-semigroup in terms of a hesitant fuzzy simple ordered Γ-semigroup.

The main aim of this article is to introduce the concept of a sup-hesitant fuzzy ideal of a ternary semigroup, which is a generalization of a hesitant fuzzy ideal and an interval-valued fuzzy ideal in a ternary semigroup. Some characterizations of an sup-hesitant fuzzy ideal are examined in terms of a fuzzy set, a hesitant fuzzy set, and an interval valued fuzzy set. Further, we discuss the relation between an ideal and a generalization of a characteristic hesitant fuzzy set and a characteristic interval-valued fuzzy set.

### 2. Preliminaries

In the following sections, we introduce some definitions and results that are important for the present study.

By a ternary semigroup, we mean a set T ≠ ∅ with a ternary operation T ×T ×TT, written as (t1, t2, t3) ? t1t2t3 satisfying the identity (for all t1, t2, t3, t4, t5T)((t1t2t3)t4t5 = t1(t2t3t4)t5 = t1t2(t3t4t5)). Throughout this paper, T is represented as a ternary semigroup. Let X ≠ ∅, Y ≠ ∅, and Z ≠ ∅ be subsets of T. We define the subset XYZ of T as follows: XYZ = {xyz | xX, yY, zZ}. A subset A ≠ ∅ of T is said to be a left (lateral, right) ideal (L(Lt, R)I) of T and TTAA (TATA, ATTA). If the subset is an LI, LtI, and RI of T, then it is said to be an ideal (Id) of T.

A fuzzy set (FS) f [4] in set X ≠ ∅ is a mapping from X to the unit segment of the real line [0, 1]. Kar and Sarkar [5] studied an FS in a ternary semigroup and introduced the concepts of a fuzzy left (lateral, right) ideal and a fuzzy ideal of ternary semigroups as follows:

### Definition 2.1 [5]

Let f be the FS in T. Then, f is said to be

• (1) a fuzzy left ideal (FLI) of T while (for all t1, t2, t3T)(f(t3) ≤ f(t1t2t3)),

• (2) a fuzzy lateral ideal (FLtI) of T while (for all t1, t2, t3T)(f(t2) ≤ f(t1t2t3)),

• (3) a fuzzy right ideal (FRI) of T while (for all t1, t2, t3T)(f(t1) ≤ f(t1t2t3)), or

• (4) a fuzzy ideal (FI) of T while f is an FLI, an FLtI, and an FRI of T, that is, (for all t1, t2, t3T)(max{f(t1), f(t2), f(t3)} ≤ f(t1t2t3)).

Let [[0, 1]] be the set of all closed subintervals of [0, 1]; that is

$[[0,1]]={[t-,t+]?t-,t+∈[0,1] and t-≤t+}.$

Let $t1^=[t1-,t1+],t2^=[t2-,t2+]∈[[0,1]]$. We define the operations ?, =, ?, and rmax as follows:

• (1) $t1^?t2^⇔t1-≤t2-,t1+≤t2+$,

• (2) $t1^=t2^⇔t1-=t2-,t1+=t2+$,

• (3) $t1^?t2^⇔t1^?t2^,t1^≠t2^$,

• (4) $rmax{t1^,t2^}=[max{t1-,t2-},max{t1+,t2+}]$.

Let X ≠ ∅ be a set. A mapping ν?: X → [[0, 1]] is said to be an interval-valued fuzzy set (IvFS) [12] on X, where for any xX, ν?(x) = [ν(x), ν+(x)], anything ν and ν+ are FSs in X such that ν (x) ≤ ν+(x).

For a subset A of X, the characteristic interval-valued fuzzy set CIA of A on X is defined by

$CIA:X→[[0,1]],x?{1^if x∈A,0^otherwise,$

where 0? = [0, 0] and 1? = [1, 1].

### Definition 2.2 [9]

Let ν? be an IvFS on T. Then, ν? is said to be

• (1) an interval-valued fuzzy left ideal (IvFLI) of T while (for all t1, t2, t3T)(ν?(t3) ? ν?(t1t2t3)),

• (2) an interval-valued fuzzy lateral ideal (IvFLtI) of T while (for all t1, t2, t3T)(ν?(t2) ? ν?(t1t2t3)),

• (3) an interval-valued fuzzy right ideal (IvFRI) of T while (for all t1, t2, t3T)(ν?(t1) ? ν?(t1t2t3)),

• (4) an interval-valued fuzzy ideal (IvFI) of T while it is an IvFLI, an IvFLtI, and an IvFRI of T, that is, (for all t1, t2, t3T)(rmax{ν?(t1), ν?(t2), ν?(t3)} ? ν?(t1t2t3)).

### Theorem 2.3 [9]

A subset A ≠ ∅ of T is an Id of T if and only if CIA is an IvFI of T.

Torra and his colleague [13,14] defined a hesitant fuzzy set (HFS) on a set X ≠ ∅ in terms of a mapping h that, when applied to X, returns a subset of [0, 1], that is, h: X → ℘[0, 1], where ℘[0, 1] denotes the set of all subsets of [0, 1]. Talee et al. [15] applied the concept of an HFS to a ternary semigroup and introduced the concepts of a hesitant fuzzy left (lateral, right) ideal and a hesitant fuzzy ideal of a ternary semigroup as follows:

### Definition 2.4 [15]

Let h be an HFS on T. Then, h is said to be

• (1) a hesitant fuzzy left ideal (HFLI) of T while (for all t1, t2, t3T)(h(t3) ⊆ h(t1t2t3)),

• (2) a hesitant fuzzy lateral ideal (HFLtI) of T while (for all t1, t2, t3T)(h(t2) ⊆ h(t1t2t3)),

• (3) a hesitant fuzzy right ideal (HFRI) of T while (for all t1, t2, t3T)(h(t1) ⊆ h(t1t2t3)),

• (4) a hesitant fuzzy ideal (HFI) of T while it is a HFLI, a HFLtI, and a HFRI of T, that is, (for all t1, t2, t3T)(h(t1) ∪ h(t2) ∪ h(t3) ⊆ h(t1t2t3)).

For a subset A of a set X ≠ ∅, define the characteristic hesitant fuzzy set (CHFS) CHA of A on X as follows:

$CHA:X→P[0,1],x?{[0,1]while x∈A,∅otherwise.$

### Theorem 2.5 [15]

A subset A ≠ ∅ of T is an Id of T if and only if CHA is an HFI of T.

It is well known that an HFS on T is a generalization of the concept of an IvFS on T. In general, we can see that the HFI of T is not an IvFI of T, and an IvFI of T is not an HFI of T, as shown in Example 2.6.

### Example 2.6

Consider a ternary semigroup T = {−i, 0, i} under the usual multiplication over a complex number.

• (1) Define an HFS h on T by h(i) = h(−i) = {0.1, 0.2, 0.3, 0.5} and h(0) = [0.1, 0.5], and we have h as an HFI of T but not an IvFI of T because h is not an IvFS on T.

• (2) Define an IvFS ν? on T by ν?(−i) = ν?(i) = [0, 0.5] and ν?(0) = [0.5, 1], and we have ν? as an IvFI of T but not an HFI of T because

$ν^(i)∪ν^(-i)∪ν^(0)=[0,1]⊄[0.5,1]=ν^(0)=ν^((i)(0)(-i)).$

• (3) Define an IvFS g on T by g(i) = g(−i) = [0, 0.4] and g(0) = [0, 1]. Then, g is both an HFI and an IvFI of T.

### 3. Main Results

For ∇ ∈ ℘[0, 1], define SUP ∇ by

$SUP∇={sup ∇while ∇≠∅,0otherwise.$

For an HFS h on X and ∇ ∈ ℘[0, 1], we define SUP [h; ∇] as

$SUP [h;∇]={x∈X?SUP h(x)≥SUP ∇}.$

### Definition 3.1

Given ∇ ∈ ℘[0, 1], an HFS h on T is said to be a sup-hesitant fuzzy left (lateral, right) ideal of T related to ∇ (∇-sup-HFL(Lt, R)I of T), whereas the set SUP [h; ∇] is an L(Lt, R)I of T. If h is a ∇-sup-HFL(Lt, R)I of T for all ∇ ∈ ℘[0, 1] when SUP [h; ∇] ≠ ∅, then h is said to be a sup-hesitant fuzzy left (lateral, right) ideal (sup-HFL(Lt, R)I) of T.

### Definition 3.2

An HFS h on T is said to be a sup-hesitant fuzzy ideal of T related to ∇ (∇-sup-HFI of T), whereas it is an ∇-sup-HFLI, a ∇-sup-HFLtI, and a ∇-sup-HFRI of T. If h is a ∇-sup-HFI of T for all ∇ ∈ ℘[0, 1] when SUP [h; ∇] ≠ ∅, then h is said to be a sup-hesitant fuzzy ideal (sup-HFI) of T.

### Lemma 3.3

All IvFL(Lt, R)Is of T are a sup-HFL(Lt, R)I.

Proof

Suppose that ν? is an IvFLI of T and ∇ ∈ ℘[0, 1] such that SUP [ν?; ∇] ≠ ∅. Let a, bT, and let c ∈ SUP [ν?; ∇]. Then, sup ν? (c) ≥ SUP ∇. Because ν? is an IvFLI of T, we have

$SUP ∇≤sup ν^(c)=ν+(c)≤ν+(abc)=sup ν^(abc).$

Thus, abc ∈ SUP [ν?; ∇]. Hence, SUP [ν?; ∇] is an LI of T, which indicates that ν? is a ∇-sup-HFLI of T. Therefore, we conclude that ν? is a sup-HFLI of T.

From Lemma 3.3, we obtain Theorem 3.4.

### Theorem 3.4

All IvFIs of T are a sup-HFI.

The converses of Lemma 3.3 and Theorem 3.4 are not true, as shown in Example 3.5.

### Example 3.5

Consider a ternary semigroup T = {O, A, B, C, D, I} under the usual matrix multiplication, where

$O=(0000),A=(1000),B=(0100),C=(0010),D=(0001),I=(1001).$

Define an IvFS ν? on T by

$ν^(O)=[0,1], ν^(A)=[0.4,1], ν^(B)=[0.6,1],ν^(C)=ν^(D)=[0.5,1], ν^(I)=0^.$

Thus, ν? is a sup-HFI of T but not an IvFI of T. Moreover, we know that

• (1) ν? is not an IvFLI of T because ν?(OAB) = [0, 1] ? [0.6, 1] = ν?(B).

• (2) ν? is not an IvFLtI of T because ν?(OAB) = [0, 1] ? [0.4, 1] = ν?(A).

• (3) ν? is not an IvFRI of T because ν?(CBO) = [0, 1] ? [0.5, 1] = ν?(C).

From Lemma 3.3, Theorem 3.4, and Example 3.5, we find that in an arbitrary ternary semigroup, a sup-HFL(Lt, R)I is a generalization of the concept of an IvFL(Lt, R)I, and a sup-HFI is a generalization of the concept of an IvFI.

### Lemma 3.6

All HFL(Lt, R)Is of T are a sup-HFL(Lt, R)I.

Proof

Suppose that h is an HFLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. Then, SUP h(c) ≥ SUP ∇. Because h is an HFLI of T, we have h(c) ⊆ h(abc) and thus SUP h(c) ≤ SUP h(abc). Therefore, abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T, which signifies that h is a ∇-sup-HFLI of T. We thus conclude that h is a sup-HFLI of T.

From Lemma 3.6, we obtain Theorem 3.7.

### Theorem 3.7

All HFIs of T are a sup-HFI.

Example 3.8 shows that the converses of Lemmas 3.6 and Theorem 3.7 do not hold.

### Example 3.8

Consider a ternary semigroup T = {O, I, X, Y, Z} under the usual matrix multiplication, where

$O=(000000000), I=(100010001), X=(000010000),Y=(001000000),Z=(000000100).$

Define an HFS h on T as

$h(O)={0,1}, h(X)=[0,1], h(Y)=h(Z)=[0,1),h(I)=∅.$

Thus, h is a sup-HFI of T, but not an HFI of T. Moreover, we know that

• (1) h is not an HFLI of T because h(X) = [0, 1] ⊃ {0, 1} = h(OYX).

• (2) h is not an HFLtI of T because h(X) = [0, 1] ⊃ {0, 1} = h(OXI).

• (3) h is not an HFRI of T because h(X) = [0, 1] ⊃ {0, 1} = h(XOZ).

From Lemma 3.6, Theorem 3.7, and Example 3.8, we find that in an arbitrary ternary semigroup, a sup-HFL(Lt, R)I is a generalization of the concept of an HFL(Lt, R)I, and a sup-HFI is a generalization of the concept of an HFI.

Let h be an HFS on T, and define the FS Fh in T as

$Fh:T→[0,1],x?SUP h(x).$

The following lemma characterizes the sup-types of HFSs on T by FS Fh.

### Lemma 3.9

An HFS h on T is a sup-HFL(Lt, R)I of T if and only if Fh is an FL(Lt, R)I of T.

Proof

Suppose that h is an sup-HFLI of T. Let a, b, cT, and let ∇ = h(c). Then, c ∈ SUP [h; ∇]. Thus, h is a ∇-sup-HFLI of T, which indicates that SUP [h; ∇] is an LI of T. Hence, abc ∈ SUP [h; ∇] and thus

$Fh(abc)=SUP h(abc)≥SUP ∇=SUP h(c)=Fh(c).$

Therefore, Fh is an FLI of T.

Conversely, suppose that Fh is an FLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. Then,

$SUP h(abc)=Fh(abc)≥Fh(c)=SUP h(c)≥SUP ∇,$

and it is implied that abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T; that is, h is a ∇-sup-HFLI of T. Therefore, we conclude that h is a sup-HFLI of T.

Let h be an HFS on T and ∇ ∈ ℘[0, 1], and we define the HFS H(h; ∇) on T as

$(for all x∈T)(H(h;∇)(x)={t∈∇?SUP h(x)≥t}).$

We then denote H(h;?xT h(x)) by Hh and H(h; [0, 1]) by Ih. Then, Ih is an IvFS on T.

### Remark 3.10

If h is an HFS on T, then h(x) ⊆ Hh(x) ⊆ Ih(x) and SUP h(x) = SUPHh(x) = supIh(x) for all xT.

Now, we study sup-types of HFSs on T using the HFS H(h; ∇) and the IvFS Ih.

### Lemma 3.11

An HFS h on T is a sup-HFL(Lt, R)I of T if and only if H(h; ∇) is a HFL(Lt, R)I of T for all ∇ ∈ ℘[0, 1].

Proof

Suppose that h is a sup-HFLI of T and ∇ ∈ ℘[0, 1]. Let a, b, cT. If H(h; ∇)(c) is empty, then H(h; ∇)(c) ⊆ H(h; ∇)(abc). In addition, let t ∈ H(h; ∇)(c). Then, t ∈ ∇, SUP h(c) ≥ t, and c ∈ SUP [h; h(c)]. Because h is a sup-HFLI of T, we have SUP [h; h(c)] as an LI of T. Hence, abc ∈ SUP [h; h(c)], which indicates that SUP h(abc) ≥ SUP h(c) ≥ t. Thus, t ∈ H(h; ∇)(abc). Therefore, H(h; ∇)(c) ⊆ H(h; ∇)(abc). Consequently, H(h; ∇) is an HFLI of T.

Conversely, suppose that H(h; ∇) is an HFLI of T for all ∇ ∈ ℘[0, 1]. Let a, b, cT and ∇ ∈ ℘[0, 1] exist such that c ∈ SUP [h; ∇]. Then, H(h; ∇)(c) = ∇, and by assumption, we have ∇ = H(h; ∇)(c) ⊆ H(h; ∇)(abc). Thus, SUP h(abc) ≥ SUP ∇, and it is implied that abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T; that is, h is a ∇-sup-HFLI of T. Therefore, we conclude that h is a sup-HFLI of T.

### Theorem 3.12

For an HFS h on T, the following statements are equivalent.

• (1) h is a sup-HFL(Lt, R)I of T.

• (2) Hh is an HFL(Lt, R)I of T.

• (3) Hh is a sup-HFL(Lt, R)I of T.

• (4) Ih is an IvFL(Lt, R)I of T.

• (5) Ih is a sup-HFL(Lt, R)I of T.

• (6) Ih is an HFL(Lt, R)I of T.

Proof

(1) ⇒ (2) and (1) ⇒ (6). These follow from Lemma 3.11.

(2) ⇒ (3) and (6) ⇒ (5). These follow from Lemma 3.6.

(4) ⇒ (5). This follows from Lemma 3.3.

(3) ⇒ (1). Suppose that Hh is an sup-HFLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. Based on Remark 3.10, we have SUPHh(c) = SUP h(c) ≥ SUP ∇ and thus c ∈ SUP [Hh; ∇]. We assume that SUP [Hh; ∇] is an LI of T, and then abc ∈ SUP [Hh; ∇]. By Remark 3.10 again, we can see that SUP h(abc) = SUPHh (abc) ≥ SUP ∇, which signifies that abc ∈ SUP [h; ∇]. Hence, SUP [h; ∇] is an LI of T; that is, h is a ∇-sup-HFLI of T. We therefore conclude that h is a sup-HFLI of T.

(1) ⇒ (4). Suppose that h is a sup-HFLI of T and a, b, cT. Then, c ∈ SUP [h; h(c)], and therefore by assumption we have abc ∈ SUP [h; h(c)]. Thus, SUP h(c) ≤ SUP h(abc), and therefore Ih(c) = [0, SUP h(c)] ? [0, SUP h(abc)] = Ih(abc). Hence, Ih is an IvFLI of T.

(5) ⇒ (1). Let Ih be an sup-HFLI of T and ∇ ∈ ℘[0, 1] such that SUP [h; ∇] ≠ ∅. Let a, bT and c ∈ SUP [h; ∇]. By Remark 3.10, we have sup Ih(c) = SUPh(c) ≥ SUP ∇, and thus c ∈ SUP [Ih; ∇]. We assume that abc ∈ SUP [Ih; ∇]. By Remark 3.10, we obtain SUP h(abc) = supIh (abc) ≥ SUP ∇, which indicates that abc ∈ SUP [h; ∇]. Hence, SUP[h; ∇] is an LI of T, which signifies that h is a ∇-sup-HFLI of T. Therefore, we conclude that h is a sup-HFLI of T.

From Lemma 3.9 and Theorem 3.12, we obtain Theorem 3.13.

### Theorem 3.13

For an HFS h on T, the following statements are equivalent.

• (1) h is a sup-HFI of T.

• (2) (for all a, b, cT)(SUP h(abc) ≥ max{SUP h(a), SUP h(b), SUP h(c)}).

• (3) Fh is an FI of T.

• (4) Hh is an HFI of T.

• (5) Hh is a sup-HFI of T.

• (6) Ih is an IvFI of T.

• (7) Ih is a sup-HFI of T.

• (8) Ih is an HFI of T.

For a subset A of T and ∇, Ω ∈ ℘[0, 1] with SUP ∇ < SUP Ω, we define a map $HA(∇,Ω)$ as follows:

$HA(∇,Ω):T→P[0,1],x?{Ωwhile x∈A,∇otherwise.$

Then, $HA(∇,Ω)$ is an HFS on T, which is said to be a sup (∇, Ω)-characteristic hesitant fuzzy set (sup (∇, Ω)-CHFS) of A of T. In addition, sup (∇, Ω)-CHFS with ∇ = ∅ and Ω = [0, 1] is the CHFS of A, that is, $HA(∅,[0,1])=CHA$. Moreover, sup (∇, Ω)-CHFS with ∇ = 0? and Ω = 1? is the CIvFS of A, that is, $HA(0^,1^)=CIA$.

### Theorem 3.14

Let a subset A ≠ ∅ of T and ∇, Ω ∈ ℘[0, 1] exist such that SUP ∇ < SUP Ω. Then, A is an Id of T if and only if $HA(∇,Ω)$ is an sup-HFI of T.

Proof

Suppose that there exist a, b, cT such that

$SUP HA(∇,Ω)(abc). Then,

$HA(∇,Ω)(a)=Ω, HA(∇,Ω)(b)=Ω$, or $HA(∇,Ω)(c)=Ω$, which signifies that aA, bA, or cA. Because A is an Id of T, we have abcA and $HA(∇,Ω)(abc)=Ω$. Thus,

$SUP HA(∇,Ω)(abc)=max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(b),SUP HA(∇,Ω)(c)}$

is a contradiction. Hence,

$SUP HA(∇,Ω)(abc)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(b),SUP HA(∇,Ω)(c)}$

for all a, b, cT, and by Theorem 3.13, we have $HA(∇,Ω)$ being a sup-HFI of T.

Conversely, let aA and x, yT. Then $HA(∇,Ω)(a)=Ω$. Because $HA(∇,Ω)$ is a sup-HFI of T, and by Theorem 3.13, we have

$SUP HA(∇,Ω)(axy)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(x),SUP HA(∇,Ω)(y)},SUP HA(∇,Ω)(xay)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(x),SUP HA(∇,Ω)(y)},$

and

$SUP HA(∇,Ω)(xya)≥max{SUP HA(∇,Ω)(a),SUP HA(∇,Ω)(x),SUP HA(∇,Ω)(y)}=SUP Ω.$

Thus,

$SUP HA(∇,Ω)(axy)=SUP HA(∇,Ω)(xay)=SUP HA(∇,Ω)(xya)=SUP Ω,$

which indicates that axy, xay, xyaA. Hence, A is the Id of T.

From Theorems 2.3, 2.5, 3.4, 3.7, and 3.14, we obtain Theorem 3.15.

### Theorem 3.15

For a subset A ≠ ∅ of T, the following statements are equivalent.

• (1) A is an Id of T.

• (2) CIA is an IvFI of T.

• (3) CIA is a sup-HFI of T.

• (4) CHA is an HFI of T.

• (5) CHA is a sup-HFI of T.

• (6) $HA(∇,Ω)$ is a sup-HFI of T for all ∇, Ω ∈ P[0, 1] with SUP ∇ < SUP Ω

### 4. Conclusion

In this paper, we introduced the concept of a sup-HFI in a ternary semigroup, which is a generalization of an HFI and an IvFI in a ternary semigroup, and examined some characterizations of a sup-HFI in terms of an FS, an HFS, and an IvFS. Further, we discussed the relation between an Id and the generalizations of CHFSs and CIvFSs. As important study results, we found that the following statements are all equivalent in a ternary semigroup T: A subset A is an Id, CIA is an IvFI, CIA is a sup-HFI, CHA is an HFI, and CHA is a sup-HFI.

In the future, we will study a sup-HFI in a Γ semigroup and examine some characterizations of a sup-HFI in terms of an FS, an HFS, and an IvFS.

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