International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 135-143
Published online June 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.2.135
© The Korean Institute of Intelligent Systems
Thiti Gaketem^{1} and Pannawit Khamrot^{2}
^{1}Department of Mathematics, School of Science, University of Phayao, Phayao, Thailand
^{2}Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok, Thailand
Correspondence to :
Thiti Gaketem (josemour25@yahoo.com)
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.
In this study, we define new types of intuitionistic fuzzy ideals and intuitionistic fuzzy almost ideals in Γ-semigroups, which can be applied to minimal ideals. We discuss the properties of intuitionistic fuzzy ideals and almost ideals in Γ-semigroups. Moreover, minimal ideals and almost minimal ideals are investigated with respect to their properties.
Keywords: Intuitionistic fuzzy ideals, Intuitionistic fuzzy almost ideals, Minimal intuitionistic fuzzy ideal, Minimal intuitionistic almost ideal
The theory for relation to the theory of fuzzy sets was first proposed by Zadeh [1]. This theory has been applied in many fields, including medical science, robotics, computer science, information science, control engineering, measure theory, logic, set theory, and topology. In 2000, the concept of intuitionistic fuzzy sets was developed by Atanassov as a generalization of the fuzzy sets, and it was used for the study of vagueness. In 2011, Sarar et al. [2] studied intuitionistic fuzzy sets in Γ-semigroups and investigated the properties of intuitionistic fuzzy ideals in Γ-semigroups. The structure of the ideal theory is important in the study of semigroups, and many researchers use the knowledge of ideals in studies related to Gamma-semigroups in fuzzy semigroups. For instance, Chinram [3] studied almost quasi-Γ-ideal and fuzzy almost quasi-Γ-ideals in Γ-semigroups; Marapureddy and Doradla [8] studied weak interior ideals of Γ-semigroups; and Majumder and Mandal [3] studied fuzzy generalized bi-ideal in Γ-semigroups. In regards to the concept of intuitionistic fuzzy ideals, several researchers expanded on this idea [4–7]. Recently, in 2021, Simuen et al. [8], Simuen et al. studied the concept of ideals and fuzzy ideals of Γ-semigroups.
In this study, we extend the concept of new fuzzy ideals to intuitionistic fuzzy ideals of Γ-semigroups and investigate the properties of the new types intuitionistic fuzzy ideals.
With respect to the content of this paper, some basic definitions are provided below, which are important for the proper understanding of the theory presented in the next section.
A
Let
(1) A
(2) An
(3) An
(4) A left
(5) An (
For any
For any
A
For any two fuzzy sets
(1)
(2)
(3) (
(4) (
For any two fuzzy sets of
For any element
For two fuzzy sets
The following conditions define the types of fuzzy almost ideal on semigroups.
A fuzzy set
(1) a
(2) a
(3) a
(4) a
The following conditions define the types of fuzzy sub-semigroups on Γ-semigroups.
A fuzzy set
(1) A
(2) A
(3) A
(4) A
Now, we review the definition of the intuitionistic fuzzy sets and their basic properties, which are discussed in the following section.
Let
which satisfies 0 ≤
For IF sets
(1)
(2)
(3)
(4)
For
For IF sets
and
For any element
For the IF sets
and
Let
and
For the sake of simplicity, we use the symbol
An IF set
(1)
(2)
(3)
(4)
for all
In this section, we define the bipolar fuzzy (
Let
(1) An
(2) An
(3) An
(4) An
Let
Let us suppose that
If
If
Conversely, assuming that
The intersection and union of any two IF left
Let
and
Thus,
and
Thus,
Next, we provide the definitions for IF (
Let
Let
Let us suppose that
If
If
Therefore,
Conversely, assuming that
The intersection and union of any two IF (
Let
and
Thus,
Next, we define conditions for IF (
Let
If
Let
and
Thus,
Every IF (
Let
Additionally, we obtain:
Similarly, we can show that:
and
Hence,
Let
Let us suppose that
If
If
Therefore,
Conversely, assuming that
Let
(1) An
(2) An
(3) An
Here,
If
Let us suppose that
Let
Let us suppose that
Conversely, assuming that
Next, we review the definition of supp(
Let
Let
Let
Conversely, let supp(
An ideal
An IF almost left
Let
Let us suppose that
Conversely, let us suppose that
Next, we provide the definitions for IF almost (
Let
If
Let us suppose that
Let
Let us suppose that
Conversely, assuming that
In the following section, we study the properties and relationship between supp(
Let
Let
Conversely, let supp(
An almost ideal
An IF almost (
Let
Let us suppose that
Conversely, let us suppose that
Next, we present the definitions of IF almost (
Let
If
Let us suppose that
Let
Let us suppose that
Conversely, assuming that
In the next section, we study properties and relationship between supp(
Let
Let
Conversely, let supp(
An IF almost (
Let
Let us suppose that
Conversely, let us suppose that
In this study, we present the concepts of intuitionistic fuzzy ideals and almost ideals in Γ-semigroups and apply the minimal condition to Γ-semigroups. In our future course of study, we plan to extend the concepts presented here to algebraic systems including, hyper semigroups, IUP-algebras, and UP-algebras.
No potential conflict of interest relevant to this study was reported.
E-mail: josemour25@yahoo.com
E-mail: pk g@rmutl.ac.th
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 135-143
Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.135
Copyright © The Korean Institute of Intelligent Systems.
Thiti Gaketem^{1} and Pannawit Khamrot^{2}
^{1}Department of Mathematics, School of Science, University of Phayao, Phayao, Thailand
^{2}Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok, Thailand
Correspondence to:Thiti Gaketem (josemour25@yahoo.com)
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.
In this study, we define new types of intuitionistic fuzzy ideals and intuitionistic fuzzy almost ideals in Γ-semigroups, which can be applied to minimal ideals. We discuss the properties of intuitionistic fuzzy ideals and almost ideals in Γ-semigroups. Moreover, minimal ideals and almost minimal ideals are investigated with respect to their properties.
Keywords: Intuitionistic fuzzy ideals, Intuitionistic fuzzy almost ideals, Minimal intuitionistic fuzzy ideal, Minimal intuitionistic almost ideal
The theory for relation to the theory of fuzzy sets was first proposed by Zadeh [1]. This theory has been applied in many fields, including medical science, robotics, computer science, information science, control engineering, measure theory, logic, set theory, and topology. In 2000, the concept of intuitionistic fuzzy sets was developed by Atanassov as a generalization of the fuzzy sets, and it was used for the study of vagueness. In 2011, Sarar et al. [2] studied intuitionistic fuzzy sets in Γ-semigroups and investigated the properties of intuitionistic fuzzy ideals in Γ-semigroups. The structure of the ideal theory is important in the study of semigroups, and many researchers use the knowledge of ideals in studies related to Gamma-semigroups in fuzzy semigroups. For instance, Chinram [3] studied almost quasi-Γ-ideal and fuzzy almost quasi-Γ-ideals in Γ-semigroups; Marapureddy and Doradla [8] studied weak interior ideals of Γ-semigroups; and Majumder and Mandal [3] studied fuzzy generalized bi-ideal in Γ-semigroups. In regards to the concept of intuitionistic fuzzy ideals, several researchers expanded on this idea [4–7]. Recently, in 2021, Simuen et al. [8], Simuen et al. studied the concept of ideals and fuzzy ideals of Γ-semigroups.
In this study, we extend the concept of new fuzzy ideals to intuitionistic fuzzy ideals of Γ-semigroups and investigate the properties of the new types intuitionistic fuzzy ideals.
With respect to the content of this paper, some basic definitions are provided below, which are important for the proper understanding of the theory presented in the next section.
A
Let
(1) A
(2) An
(3) An
(4) A left
(5) An (
For any
For any
A
For any two fuzzy sets
(1)
(2)
(3) (
(4) (
For any two fuzzy sets of
For any element
For two fuzzy sets
The following conditions define the types of fuzzy almost ideal on semigroups.
A fuzzy set
(1) a
(2) a
(3) a
(4) a
The following conditions define the types of fuzzy sub-semigroups on Γ-semigroups.
A fuzzy set
(1) A
(2) A
(3) A
(4) A
Now, we review the definition of the intuitionistic fuzzy sets and their basic properties, which are discussed in the following section.
Let
which satisfies 0 ≤
For IF sets
(1)
(2)
(3)
(4)
For
For IF sets
and
For any element
For the IF sets
and
Let
and
For the sake of simplicity, we use the symbol
An IF set
(1)
(2)
(3)
(4)
for all
In this section, we define the bipolar fuzzy (
Let
(1) An
(2) An
(3) An
(4) An
Let
Let us suppose that
If
If
Conversely, assuming that
The intersection and union of any two IF left
Let
and
Thus,
and
Thus,
Next, we provide the definitions for IF (
Let
Let
Let us suppose that
If
If
Therefore,
Conversely, assuming that
The intersection and union of any two IF (
Let
and
Thus,
Next, we define conditions for IF (
Let
If
Let
and
Thus,
Every IF (
Let
Additionally, we obtain:
Similarly, we can show that:
and
Hence,
Let
Let us suppose that
If
If
Therefore,
Conversely, assuming that
Let
(1) An
(2) An
(3) An
Here,
If
Let us suppose that
Let
Let us suppose that
Conversely, assuming that
Next, we review the definition of supp(
Let
Let
Let
Conversely, let supp(
An ideal
An IF almost left
Let
Let us suppose that
Conversely, let us suppose that
Next, we provide the definitions for IF almost (
Let
If
Let us suppose that
Let
Let us suppose that
Conversely, assuming that
In the following section, we study the properties and relationship between supp(
Let
Let
Conversely, let supp(
An almost ideal
An IF almost (
Let
Let us suppose that
Conversely, let us suppose that
Next, we present the definitions of IF almost (
Let
If
Let us suppose that
Let
Let us suppose that
Conversely, assuming that
In the next section, we study properties and relationship between supp(
Let
Let
Conversely, let supp(
An IF almost (
Let
Let us suppose that
Conversely, let us suppose that
In this study, we present the concepts of intuitionistic fuzzy ideals and almost ideals in Γ-semigroups and apply the minimal condition to Γ-semigroups. In our future course of study, we plan to extend the concepts presented here to algebraic systems including, hyper semigroups, IUP-algebras, and UP-algebras.