International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(4): 290-297
Published online December 25, 2020
https://doi.org/10.5391/IJFIS.2020.20.4.290
© The Korean Institute of Intelligent Systems
Warud Nakkhasen
Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, Thailand
Correspondence to :
Warud Nakkhasen (warud.n@msu.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.
We define the concept of intuitionistic fuzzy ideals of ternary near-rings as a generalization of fuzzy ideals, and we investigate some of their properties. Moreover, we characterize the notions of Noetherian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.
Keywords: Ternary near-ring, Fuzzy ideal, Intuitionistic fuzzy ideal, Intuitionistic fuzzy set
The concept of fuzzy sets was first introduced by Zadeh [1] as a function of a nonempty set
The concept of intuitionistic fuzzy sets was introduced by Atanassov [11–13] as a generalization of the concept of fuzzy sets. Fuzzy sets give the degree of membership of an element in a given set. Intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership. This theory has been studied by many mathematicians [14–19]. Biswas [20] considered the notion of intuitionistic fuzzy subgroups of groups. In [21], the authors presented the concept of intuitionistic fuzzy ideals of semi-rings. Later, the concept of intuitionistic fuzzy ideals of near-rings was introduced and studied by Zhan and Ma [22].
In 2012, Nakkhasen and Pibaljomme [23] introduced the concept of left ternary near-rings and investigated some properties of
In this section, we present the basic definitions that are used in the following sections of this paper.
A
Let
Let
(i) (
(ii) (
(iii) [
A nonempty subset
Let
(i)
(ii)
(iii)
We call
Let
+ | ||||
---|---|---|---|---|
The ternary operation [ ] on
Let
Let
The concept of intuitionistic fuzzy sets was introduced by Atanassov [11–13] as an important generalization of the concept of fuzzy sets. An
where
In this section, we introduce the concept of intuitionistic fuzzy ideals of ternary near-rings and investigate some of their properties.
An intuitionistic fuzzy set
(IF1)
(IF2)
(IF3)
(IF4)
(IF5)
(IF6)
(AF1)
(AF2)
(AF3)
(AF4)
(AF5)
(AF6)
In Example 1, we define an intuitionistic fuzzy set
If
Assume that
Let
Assume that
Conversely, assume that for any for any
In Example 2, it is clear that
Let
Assume that
Conversely, assume that
The following theorem immediately follows from Theorem 2.
Let
If
Assume that
The proofs of (IF2)–(IF6) are similar to that of (IF1). Next, (AF1) we have
The proofs of (AF2)–(AF6) are similar to that of (AF1). Therefore,
If
The proof is similar to Theorem 4.
Let
Let
Assume that
If we strengthen the condition of
Let
Assume that
Let
Let
Assume that
Conversely, assume that
Let
Let
Assume that
Conversely, assume that
as
Let
Let
In this section, we define the notions of Noetherian and Artinian ternary near-rings and characterize Northerian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.
A ternary near-ring
there exists
If every intuitionistic fuzzy ideal of a ternary near-ring
Assume that every intuitionistic fuzzy ideal of a ternary near-ring
for all
The proof of the following theorem is similar to that of Theorem 11.
If every intuitionistic fuzzy ideal of a ternary near-ring
A ternary near-ring
Assume that
Conversely, suppose that
for all
A ternary near-ring
Suppose that
Let
Let
Conversely, it follows by Theorem 11 and Theorem 12.
We introduced the concept of the intuitionistic fuzzy ideal in ternary near-rings as a generalization of their fuzzy ideals and studied some of their properties. We also presented the notions of Noetherian and Artinian ternary near-rings and characterized some of their properties using their intuitionistic fuzzy ideals. In the future, we would like to investigate some of the basic properties of the concepts of fuzzy quasi-ideals and fuzzy bi-ideals in ternary near-rings. Next, we will study the concepts of intuitionistic fuzzy quasi-ideals and intuitionistic fuzzy bi-ideals in ternary near-rings as generalizations of their fuzzy quasi-ideals and fuzzy bi-ideals, respectively.
No potential conflict of interest relevant to this article is reported.
E-mail: warud.n@msu.ac.th
International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(4): 290-297
Published online December 25, 2020 https://doi.org/10.5391/IJFIS.2020.20.4.290
Copyright © The Korean Institute of Intelligent Systems.
Warud Nakkhasen
Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, Thailand
Correspondence to:Warud Nakkhasen (warud.n@msu.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.
We define the concept of intuitionistic fuzzy ideals of ternary near-rings as a generalization of fuzzy ideals, and we investigate some of their properties. Moreover, we characterize the notions of Noetherian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.
Keywords: Ternary near-ring, Fuzzy ideal, Intuitionistic fuzzy ideal, Intuitionistic fuzzy set
The concept of fuzzy sets was first introduced by Zadeh [1] as a function of a nonempty set
The concept of intuitionistic fuzzy sets was introduced by Atanassov [11–13] as a generalization of the concept of fuzzy sets. Fuzzy sets give the degree of membership of an element in a given set. Intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership. This theory has been studied by many mathematicians [14–19]. Biswas [20] considered the notion of intuitionistic fuzzy subgroups of groups. In [21], the authors presented the concept of intuitionistic fuzzy ideals of semi-rings. Later, the concept of intuitionistic fuzzy ideals of near-rings was introduced and studied by Zhan and Ma [22].
In 2012, Nakkhasen and Pibaljomme [23] introduced the concept of left ternary near-rings and investigated some properties of
In this section, we present the basic definitions that are used in the following sections of this paper.
A
Let
Let
(i) (
(ii) (
(iii) [
A nonempty subset
Let
(i)
(ii)
(iii)
We call
Let
+ | ||||
---|---|---|---|---|
The ternary operation [ ] on
Let
Let
The concept of intuitionistic fuzzy sets was introduced by Atanassov [11–13] as an important generalization of the concept of fuzzy sets. An
where
In this section, we introduce the concept of intuitionistic fuzzy ideals of ternary near-rings and investigate some of their properties.
An intuitionistic fuzzy set
(IF1)
(IF2)
(IF3)
(IF4)
(IF5)
(IF6)
(AF1)
(AF2)
(AF3)
(AF4)
(AF5)
(AF6)
In Example 1, we define an intuitionistic fuzzy set
If
Assume that
Let
Assume that
Conversely, assume that for any for any
In Example 2, it is clear that
Let
Assume that
Conversely, assume that
The following theorem immediately follows from Theorem 2.
Let
If
Assume that
The proofs of (IF2)–(IF6) are similar to that of (IF1). Next, (AF1) we have
The proofs of (AF2)–(AF6) are similar to that of (AF1). Therefore,
If
The proof is similar to Theorem 4.
Let
Let
Assume that
If we strengthen the condition of
Let
Assume that
Let
Let
Assume that
Conversely, assume that
Let
Let
Assume that
Conversely, assume that
as
Let
Let
In this section, we define the notions of Noetherian and Artinian ternary near-rings and characterize Northerian and Artinian ternary near-rings using their intuitionistic fuzzy ideals.
A ternary near-ring
there exists
If every intuitionistic fuzzy ideal of a ternary near-ring
Assume that every intuitionistic fuzzy ideal of a ternary near-ring
for all
The proof of the following theorem is similar to that of Theorem 11.
If every intuitionistic fuzzy ideal of a ternary near-ring
A ternary near-ring
Assume that
Conversely, suppose that
for all
A ternary near-ring
Suppose that
Let
Let
Conversely, it follows by Theorem 11 and Theorem 12.
We introduced the concept of the intuitionistic fuzzy ideal in ternary near-rings as a generalization of their fuzzy ideals and studied some of their properties. We also presented the notions of Noetherian and Artinian ternary near-rings and characterized some of their properties using their intuitionistic fuzzy ideals. In the future, we would like to investigate some of the basic properties of the concepts of fuzzy quasi-ideals and fuzzy bi-ideals in ternary near-rings. Next, we will study the concepts of intuitionistic fuzzy quasi-ideals and intuitionistic fuzzy bi-ideals in ternary near-rings as generalizations of their fuzzy quasi-ideals and fuzzy bi-ideals, respectively.
+ | ||||
---|---|---|---|---|
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