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
Pongpun Julatha^{1} and Aiyared Iampan^{2}
^{1}Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand
^{2}Fuzzy 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)
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.
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
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 (
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.
In the following sections, we introduce some definitions and results that are important for the present study.
By a
A
Let
(1) a
(2) a
(3) a
(4) a
Let [[0, 1]] be the set of all closed subintervals of [0, 1]; that is
Let
(1)
(2)
(3)
(4)
Let
For a subset
where 0? = [0, 0] and 1? = [1, 1].
Let
(1) an
(2) an
(3) an
(4) an
A subset
Torra and his colleague [13,14] defined a
Let
(1) a
(2) a
(3) a
(4) a
For a subset
A subset
It is well known that an HFS on
Consider a ternary semigroup
(1) Define an HFS
(2) Define an IvFS
(3) Define an IvFS
For ∇ ∈ ℘[0, 1], define SUP ∇ by
For an HFS
Given ∇ ∈ ℘[0, 1], an HFS
An HFS
All IvFL(Lt, R)Is of
Suppose that
Thus,
From Lemma 3.3, we obtain Theorem 3.4.
All IvFIs of
The converses of Lemma 3.3 and Theorem 3.4 are not true, as shown in Example 3.5.
Consider a ternary semigroup
Define an IvFS
Thus,
(1)
(2)
(3)
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.
All HFL(Lt, R)Is of
Suppose that
From Lemma 3.6, we obtain Theorem 3.7.
All HFIs of
Example 3.8 shows that the converses of Lemmas 3.6 and Theorem 3.7 do not hold.
Consider a ternary semigroup
Define an HFS
Thus,
(1)
(2)
(3)
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
The following lemma characterizes the sup-types of HFSs on
An HFS
Suppose that
Therefore, F
Conversely, suppose that F
and it is implied that
Let
We then denote H(
If
Now, we study sup-types of HFSs on
An HFS
Suppose that
Conversely, suppose that H(
For an HFS
(1)
(2) H
(3) H
(4) I
(5) I
(6) I
(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 H
(1) ⇒ (4). Suppose that
(5) ⇒ (1). Let I
From Lemma 3.9 and Theorem 3.12, we obtain Theorem 3.13.
For an HFS
(1)
(2) (for all
(3) F
(4) H
(5) H
(6) I
(7) I
(8) I
For a subset
Then,
Let a subset
Suppose that there exist
is a contradiction. Hence,
for all
Conversely, let
and
Thus,
which indicates that
From Theorems 2.3, 2.5, 3.4, 3.7, and 3.14, we obtain Theorem 3.15.
For a subset
(1)
(2) CI
(3) CI
(4) CH
(5) CH
(6)
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
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.
No potential conflict of interest relevant to this article was reported.
E-mail: pongpun.j@psru.ac.th
E-mail: aiyared.ia@up.ac.th
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.
Pongpun Julatha^{1} and Aiyared Iampan^{2}
^{1}Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand
^{2}Fuzzy 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)
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.
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
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 (
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.
In the following sections, we introduce some definitions and results that are important for the present study.
By a
A
Let
(1) a
(2) a
(3) a
(4) a
Let [[0, 1]] be the set of all closed subintervals of [0, 1]; that is
Let
(1)
(2)
(3)
(4)
Let
For a subset
where 0? = [0, 0] and 1? = [1, 1].
Let
(1) an
(2) an
(3) an
(4) an
A subset
Torra and his colleague [13,14] defined a
Let
(1) a
(2) a
(3) a
(4) a
For a subset
A subset
It is well known that an HFS on
Consider a ternary semigroup
(1) Define an HFS
(2) Define an IvFS
(3) Define an IvFS
For ∇ ∈ ℘[0, 1], define SUP ∇ by
For an HFS
Given ∇ ∈ ℘[0, 1], an HFS
An HFS
All IvFL(Lt, R)Is of
Suppose that
Thus,
From Lemma 3.3, we obtain Theorem 3.4.
All IvFIs of
The converses of Lemma 3.3 and Theorem 3.4 are not true, as shown in Example 3.5.
Consider a ternary semigroup
Define an IvFS
Thus,
(1)
(2)
(3)
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.
All HFL(Lt, R)Is of
Suppose that
From Lemma 3.6, we obtain Theorem 3.7.
All HFIs of
Example 3.8 shows that the converses of Lemmas 3.6 and Theorem 3.7 do not hold.
Consider a ternary semigroup
Define an HFS
Thus,
(1)
(2)
(3)
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
The following lemma characterizes the sup-types of HFSs on
An HFS
Suppose that
Therefore, F
Conversely, suppose that F
and it is implied that
Let
We then denote H(
If
Now, we study sup-types of HFSs on
An HFS
Suppose that
Conversely, suppose that H(
For an HFS
(1)
(2) H
(3) H
(4) I
(5) I
(6) I
(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 H
(1) ⇒ (4). Suppose that
(5) ⇒ (1). Let I
From Lemma 3.9 and Theorem 3.12, we obtain Theorem 3.13.
For an HFS
(1)
(2) (for all
(3) F
(4) H
(5) H
(6) I
(7) I
(8) I
For a subset
Then,
Let a subset
Suppose that there exist
is a contradiction. Hence,
for all
Conversely, let
and
Thus,
which indicates that
From Theorems 2.3, 2.5, 3.4, 3.7, and 3.14, we obtain Theorem 3.15.
For a subset
(1)
(2) CI
(3) CI
(4) CH
(5) CH
(6)
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
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.