International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 259-268
Published online September 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.3.259
© The Korean Institute of Intelligent Systems
Mani Parimala^{1}, Intan Muchtadi-Alamsyah^{2}, and Madeleine Al-Tahan^{3}
^{1}Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India
^{2}AlgebraResearch Division and Center of Artificial Intelligence, Institut Teknologi Bandung, Bandung, Indonesia
^{3}Department of Mathematics, Lebanese International University, Beirut, Lebanon
Correspondence to :
Intan Muchtadi-Alamsyah (ntan@math.itb.ac.id)
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 managing vagueness and complexity, the theory of picture fuzzy sets has important advantages. In addition, picture fuzzy data are useful for modeling the uncertain existence of subjective decisions and for more flexibly assessing the fuzziness and imprecision. The goal of this study is to establish and discuss their desirable properties by defining the picture fuzzy ideals of a near-ring. Based on new definitions, this paper presents some examples and properties of picture fuzzy subsets of such rings.
Keywords: Near-ring, Picture fuzzy set, Picture fuzzy ideals, Homomorphism of the near-rings
The definition of a fuzzy set with a membership function
The definition of the theory of fuzzy sets seems to be inconclusive because of the absence of a non-membership function and disregard for the margin of hesitation possibility. These shortcomings were critically analyzed by Atanassov’s intuitionistic fuzzy set (IFS) [3] including one more component called the non-membership function into the fuzzy set. The author’s theory has gained extensive recognition as an extremely valuable tool in the fields of science, technology, engineering, and medicine. One of the extensions of IFS is the picture fuzzy set (PFS), which was developed by Cuong [4].
In PFS theory, there are three components: positive, abstinence (neutral), and negative terms, and the sum of the three components is less than or equal to 1. Simultaneously, the PFS has been applied in the fields of science and engineering. The development of the PFS can be found in many fields [5–12].
In real life, there are more than one opinion for every decision. For instance, in a democratic voting system, 10,000 individuals might vote in an election. If the election commission distributes 10,000 paper ballots, each voter can only use one ballot to cast his or her vote. The election results are generally divided into four categories based on the number of paper ballots cast: “vote for the candidate (5,000),” “vote against the candidate (2,000),” “abstain from voting (2,000),” and “refuse to vote (1,000).” “Abstain from voting” indicates that the paper ballot remains blank, which contradicts both “vote for the candidate” and “vote against candidate” but does consider the possibility of voting. However, “refuse to vote” means bypassing the vote altogether. Neutrality is related to “vote for the candidate” and “vote against candidate.” A refusal is related to bypassing the vote (that is, “refuse to vote”). PFS deals with these types of cases.
The advantages and limitations of PFS are as follows:
Advantages:
The human opinions including yes, abstain, no, and refusal can be expressed.
The PFS can be transformed into IFS and FS.
Limitations:
It cannot express the evaluation information when decision makers have difficulty determining an accurate value of each membership level.
Abou-zaid [13] studied the concept of fuzzy sub-near-rings and fuzzy ideals of near-rings. Davvaz [14] proposed the notion of fuzzy ideals of near-rings with interval-value membership functions in 2001. Several studies [15–17] further explored this notion.
The notion of anti-fuzzy subgroups of groups was introduced by Biswas in [18], and the concept of anti-fuzzy R-subgroups of the near-ring was studied by Jun et al. [19]. In addition, Kim and Jun [20] introduced the notion of anti-fuzzy ideals in near-rings in an intuitionistic fuzzy environment and studied the notion of generalized anti-fuzzy bi-ideals in ordered semigroups. Several researchers have also studied the ring concept in terms of intuitionistic fuzzy environments [21,22].
The concept of fuzzy group theory was the first to be applied to many algebraic structures, and was introduced by Rosenfeld [23]. Later, the fuzzy ideals of rings were introduced and studied by Lui [24], and many researchers [25–29] have extended these concepts to the extensions of fuzzy sets. The concepts of near-rings and their properties are interesting in the field of algebraic structures. The concept of PFS theory specializes in dealing with the uncertainty parameter (neutrality), which is not found in fuzzy or IFS theories. Asif et al. [30] introduced and studied the basic concept of the picture fuzzy ideals of near-rings. This motivated us to introduce and investigate more properties of the picture fuzzy ideals of near-rings.
This study is concerned about picture fuzzy ideals of near-rings and is constructed as follows: in Section 2, we present some definitions that are used throughout the paper. In Section 3, we introduce and study the notion of picture fuzzy ideals of a near-ring and discuss the essential characteristics of near-ring picture fuzzy ideals. Moreover, we studied picture fuzzy ideals under different operations and near-ring homomorphisms.
We include some descriptions, comments, and findings in this section that are important and are regularly used throughout the paper.
A description of the picture fuzzy structure was introduced by Cuong [4] as follows:
Let be a non-empty set. Then, a picture fuzzy set (PFS) on is defined as
where are called positive, abstinence, and negative membership functions, respectively. As a condition for a PFS, , and the degree of refusal is defined as . A picture fuzzy number is defined as a triplet .
Let and be two PFSs described on , which is the universe of discourse. Then
the union of and is
the intersection of and is
the symmetric difference of and is
where
if and only if
and
The PFS-set is not a set, but is a particular unit of certain components of the set and can therefore be positioned as a set of ordered pairs: , where , and are the positive, abstinence, and negative membership functions, respectively. If .
A near-ring
(i) (
(ii) (
(iii) , for all .
To be precise, because
A non-empty set of a near-ring
(i) ( , +) is a normal subgroup of (
(ii) ; and
(iii) for all and all .
If satisfies (i) and (ii) of Definition 5, then is a left ideal of
A fuzzy set , where
(i) ,
(ii) , for all
A fuzzy set , where
(i) ;
(ii) ;
(iii) ;
(iv) for all .
Let be a fuzzy set. If it satisfies (i)–(iii), then it is said to be a fuzzy left ideal of
A fuzzy set , where
(i) ;
(ii) ;
(iii) ;
(iv) for all .
In this section, we present new concepts related to the picture fuzzy subnear-rings. In particular, we define and study the picture fuzzy ideals of the near-rings.
A picture fuzzy set in a near-ring
(i) ; and
(ii) .
Let
(i) ;
(ii) ;
(iii) ; and
(iv)
A picture fuzzy subset is said to be a picture fuzzy left ideal of
Let (
Let
(i) for all ; and
(ii) for all .
(1) Let be a picture fuzzy subnear-ring (left ideal, right ideal, or ideal) of
(2) Substituting in Condition (i) of Definition 9, we obtain for all . Because , it follows that , and for all . This completes the proof.
Let (
Then, is a picture fuzzy ideal of
Let
+_{1} | 0 | 1 | 2 | 3 |
0 | 0 | 1 | 2 | 3 |
1 | 1 | 2 | 3 | 0 |
2 | 2 | 3 | 0 | 1 |
3 | 3 | 0 | 1 | 2 |
and
·_{1} | 0 | 1 | 2 | 3 |
0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 |
2 | 0 | 1 | 2 | 3 |
3 | 0 | 1 | 2 | 3 |
Then, (
Let ℤ be the set of integers. Then, (ℤ, +, ·) is a near-ring. Here, “+” is a standard addition and “·” is defined as
and
PFS on ℤ is as follows: For all
and
It can be clearly seen that the picture fuzzy sets are picture fuzzy ideals of ℤ.
Let
+_{2} | ||||
and
·_{2} | ||||
Then, (
In this subsection, we study the different types of operations on picture fuzzy ideals under near-rings.
If and are any two picture fuzzy ideals of
Let and be two picture fuzzy ideals of
(i) .
(ii) .
(iii) .
(iv) .
Therefore, is a picture fuzzy ideal of
If are picture fuzzy ideals of
The proof follows from Theorem 1 and applies induction.
Consider (ℤ
and
Using Theorem 1, we find that the picture fuzzy set is a picture fuzzy ideal of ℤ.
The union of the picture fuzzy ideals of a near-ring
We illustrate Remark 3 through the following example.
Consider (ℤ, +, ·) as a near-ring, , and be picture fuzzy sets and their corresponding picture fuzzy ideals presented in Example 2. Then, the picture fuzzy set on ℤ is not a picture fuzzy ideal of ℤ. This can be easily observed as .
Let and be two picture fuzzy subsets of two near-rings
Similarly, we can define the direct product of
Let and be two picture fuzzy subsets of two near-rings,
(i) .
(ii) .
(iii) .
(iv) .
Let and be two picture fuzzy ideals of two near-rings
Let and be two picture fuzzy ideals of
Let .
(i) .
(ii) .
(iii) .
(iv) .
Let
The proof follows from Theorem 2 and through induction.
Let (
and
we obtain through Theorem 2 that is a picture fuzzy ideal of
We study picture fuzzy ideals under near-ring homomorphisms.
A map , where
Let be a map, where
Let be a homomorphism, where
Because is a homomorphism, it follows that for any , and , we obtain the following.
(i)
(ii)
(iii)
(iv)
Let (ℤ, +, ·), be the near-ring and picture fuzzy sets on ℤ presented in Example 1. Let (2ℤ, +, ·) be the near-ring under a standard addition of even integers and “·” be defined as
This paper contributes to the study of picture fuzzy algebraic structures by discussing the picture fuzzy ideals of near-rings. Several interesting properties, propositions, and theorems were investigated, and some examples are shown through a picture fuzzy environment. Because picture fuzzy sets are a generalization of intuitionistic fuzzy and fuzzy sets, the results in this study can be considered a generalization of picture fuzzy near-rings.
As future research, it will be interesting to study picture fuzzy quasi-ideals, picture fuzzy bi-ideals in near-rings, rough picture fuzzy near-rings, and soft picture fuzzy near-rings. We can also elaborate more results on picture fuzzy ideals for a finite number of near-rings.
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 259-268
Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.259
Copyright © The Korean Institute of Intelligent Systems.
Mani Parimala^{1}, Intan Muchtadi-Alamsyah^{2}, and Madeleine Al-Tahan^{3}
^{1}Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India
^{2}AlgebraResearch Division and Center of Artificial Intelligence, Institut Teknologi Bandung, Bandung, Indonesia
^{3}Department of Mathematics, Lebanese International University, Beirut, Lebanon
Correspondence to:Intan Muchtadi-Alamsyah (ntan@math.itb.ac.id)
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 managing vagueness and complexity, the theory of picture fuzzy sets has important advantages. In addition, picture fuzzy data are useful for modeling the uncertain existence of subjective decisions and for more flexibly assessing the fuzziness and imprecision. The goal of this study is to establish and discuss their desirable properties by defining the picture fuzzy ideals of a near-ring. Based on new definitions, this paper presents some examples and properties of picture fuzzy subsets of such rings.
Keywords: Near-ring, Picture fuzzy set, Picture fuzzy ideals, Homomorphism of the near-rings
The definition of a fuzzy set with a membership function
The definition of the theory of fuzzy sets seems to be inconclusive because of the absence of a non-membership function and disregard for the margin of hesitation possibility. These shortcomings were critically analyzed by Atanassov’s intuitionistic fuzzy set (IFS) [3] including one more component called the non-membership function into the fuzzy set. The author’s theory has gained extensive recognition as an extremely valuable tool in the fields of science, technology, engineering, and medicine. One of the extensions of IFS is the picture fuzzy set (PFS), which was developed by Cuong [4].
In PFS theory, there are three components: positive, abstinence (neutral), and negative terms, and the sum of the three components is less than or equal to 1. Simultaneously, the PFS has been applied in the fields of science and engineering. The development of the PFS can be found in many fields [5–12].
In real life, there are more than one opinion for every decision. For instance, in a democratic voting system, 10,000 individuals might vote in an election. If the election commission distributes 10,000 paper ballots, each voter can only use one ballot to cast his or her vote. The election results are generally divided into four categories based on the number of paper ballots cast: “vote for the candidate (5,000),” “vote against the candidate (2,000),” “abstain from voting (2,000),” and “refuse to vote (1,000).” “Abstain from voting” indicates that the paper ballot remains blank, which contradicts both “vote for the candidate” and “vote against candidate” but does consider the possibility of voting. However, “refuse to vote” means bypassing the vote altogether. Neutrality is related to “vote for the candidate” and “vote against candidate.” A refusal is related to bypassing the vote (that is, “refuse to vote”). PFS deals with these types of cases.
The advantages and limitations of PFS are as follows:
Advantages:
The human opinions including yes, abstain, no, and refusal can be expressed.
The PFS can be transformed into IFS and FS.
Limitations:
It cannot express the evaluation information when decision makers have difficulty determining an accurate value of each membership level.
Abou-zaid [13] studied the concept of fuzzy sub-near-rings and fuzzy ideals of near-rings. Davvaz [14] proposed the notion of fuzzy ideals of near-rings with interval-value membership functions in 2001. Several studies [15–17] further explored this notion.
The notion of anti-fuzzy subgroups of groups was introduced by Biswas in [18], and the concept of anti-fuzzy R-subgroups of the near-ring was studied by Jun et al. [19]. In addition, Kim and Jun [20] introduced the notion of anti-fuzzy ideals in near-rings in an intuitionistic fuzzy environment and studied the notion of generalized anti-fuzzy bi-ideals in ordered semigroups. Several researchers have also studied the ring concept in terms of intuitionistic fuzzy environments [21,22].
The concept of fuzzy group theory was the first to be applied to many algebraic structures, and was introduced by Rosenfeld [23]. Later, the fuzzy ideals of rings were introduced and studied by Lui [24], and many researchers [25–29] have extended these concepts to the extensions of fuzzy sets. The concepts of near-rings and their properties are interesting in the field of algebraic structures. The concept of PFS theory specializes in dealing with the uncertainty parameter (neutrality), which is not found in fuzzy or IFS theories. Asif et al. [30] introduced and studied the basic concept of the picture fuzzy ideals of near-rings. This motivated us to introduce and investigate more properties of the picture fuzzy ideals of near-rings.
This study is concerned about picture fuzzy ideals of near-rings and is constructed as follows: in Section 2, we present some definitions that are used throughout the paper. In Section 3, we introduce and study the notion of picture fuzzy ideals of a near-ring and discuss the essential characteristics of near-ring picture fuzzy ideals. Moreover, we studied picture fuzzy ideals under different operations and near-ring homomorphisms.
We include some descriptions, comments, and findings in this section that are important and are regularly used throughout the paper.
A description of the picture fuzzy structure was introduced by Cuong [4] as follows:
Let be a non-empty set. Then, a picture fuzzy set (PFS) on is defined as
where are called positive, abstinence, and negative membership functions, respectively. As a condition for a PFS, , and the degree of refusal is defined as . A picture fuzzy number is defined as a triplet .
Let and be two PFSs described on , which is the universe of discourse. Then
the union of and is
the intersection of and is
the symmetric difference of and is
where
if and only if
and
The PFS-set is not a set, but is a particular unit of certain components of the set and can therefore be positioned as a set of ordered pairs: , where , and are the positive, abstinence, and negative membership functions, respectively. If .
A near-ring
(i) (
(ii) (
(iii) , for all .
To be precise, because
A non-empty set of a near-ring
(i) ( , +) is a normal subgroup of (
(ii) ; and
(iii) for all and all .
If satisfies (i) and (ii) of Definition 5, then is a left ideal of
A fuzzy set , where
(i) ,
(ii) , for all
A fuzzy set , where
(i) ;
(ii) ;
(iii) ;
(iv) for all .
Let be a fuzzy set. If it satisfies (i)–(iii), then it is said to be a fuzzy left ideal of
A fuzzy set , where
(i) ;
(ii) ;
(iii) ;
(iv) for all .
In this section, we present new concepts related to the picture fuzzy subnear-rings. In particular, we define and study the picture fuzzy ideals of the near-rings.
A picture fuzzy set in a near-ring
(i) ; and
(ii) .
Let
(i) ;
(ii) ;
(iii) ; and
(iv)
A picture fuzzy subset is said to be a picture fuzzy left ideal of
Let (
Let
(i) for all ; and
(ii) for all .
(1) Let be a picture fuzzy subnear-ring (left ideal, right ideal, or ideal) of
(2) Substituting in Condition (i) of Definition 9, we obtain for all . Because , it follows that , and for all . This completes the proof.
Let (
Then, is a picture fuzzy ideal of
Let
+_{1} | 0 | 1 | 2 | 3 |
0 | 0 | 1 | 2 | 3 |
1 | 1 | 2 | 3 | 0 |
2 | 2 | 3 | 0 | 1 |
3 | 3 | 0 | 1 | 2 |
and
·_{1} | 0 | 1 | 2 | 3 |
0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 |
2 | 0 | 1 | 2 | 3 |
3 | 0 | 1 | 2 | 3 |
Then, (
Let ℤ be the set of integers. Then, (ℤ, +, ·) is a near-ring. Here, “+” is a standard addition and “·” is defined as
and
PFS on ℤ is as follows: For all
and
It can be clearly seen that the picture fuzzy sets are picture fuzzy ideals of ℤ.
Let
+_{2} | ||||
and
·_{2} | ||||
Then, (
In this subsection, we study the different types of operations on picture fuzzy ideals under near-rings.
If and are any two picture fuzzy ideals of
Let and be two picture fuzzy ideals of
(i) .
(ii) .
(iii) .
(iv) .
Therefore, is a picture fuzzy ideal of
If are picture fuzzy ideals of
The proof follows from Theorem 1 and applies induction.
Consider (ℤ
and
Using Theorem 1, we find that the picture fuzzy set is a picture fuzzy ideal of ℤ.
The union of the picture fuzzy ideals of a near-ring
We illustrate Remark 3 through the following example.
Consider (ℤ, +, ·) as a near-ring, , and be picture fuzzy sets and their corresponding picture fuzzy ideals presented in Example 2. Then, the picture fuzzy set on ℤ is not a picture fuzzy ideal of ℤ. This can be easily observed as .
Let and be two picture fuzzy subsets of two near-rings
Similarly, we can define the direct product of
Let and be two picture fuzzy subsets of two near-rings,
(i) .
(ii) .
(iii) .
(iv) .
Let and be two picture fuzzy ideals of two near-rings
Let and be two picture fuzzy ideals of
Let .
(i) .
(ii) .
(iii) .
(iv) .
Let
The proof follows from Theorem 2 and through induction.
Let (
and
we obtain through Theorem 2 that is a picture fuzzy ideal of
We study picture fuzzy ideals under near-ring homomorphisms.
A map , where
Let be a map, where
Let be a homomorphism, where
Because is a homomorphism, it follows that for any , and , we obtain the following.
(i)
(ii)
(iii)
(iv)
Let (ℤ, +, ·), be the near-ring and picture fuzzy sets on ℤ presented in Example 1. Let (2ℤ, +, ·) be the near-ring under a standard addition of even integers and “·” be defined as
This paper contributes to the study of picture fuzzy algebraic structures by discussing the picture fuzzy ideals of near-rings. Several interesting properties, propositions, and theorems were investigated, and some examples are shown through a picture fuzzy environment. Because picture fuzzy sets are a generalization of intuitionistic fuzzy and fuzzy sets, the results in this study can be considered a generalization of picture fuzzy near-rings.
As future research, it will be interesting to study picture fuzzy quasi-ideals, picture fuzzy bi-ideals in near-rings, rough picture fuzzy near-rings, and soft picture fuzzy near-rings. We can also elaborate more results on picture fuzzy ideals for a finite number of near-rings.
+_{1} | 0 | 1 | 2 | 3 |
0 | 0 | 1 | 2 | 3 |
1 | 1 | 2 | 3 | 0 |
2 | 2 | 3 | 0 | 1 |
3 | 3 | 0 | 1 | 2 |
·_{1} | 0 | 1 | 2 | 3 |
0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 |
2 | 0 | 1 | 2 | 3 |
3 | 0 | 1 | 2 | 3 |
+_{2} | ||||
·_{2} | ||||