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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

On Picture Fuzzy Ideals of Near-Rings

Mani Parimala1, Intan Muchtadi-Alamsyah2, and Madeleine Al-Tahan3

1Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India
2AlgebraResearch Division and Center of Artificial Intelligence, Institut Teknologi Bandung, Bandung, Indonesia
3Department of Mathematics, Lebanese International University, Beirut, Lebanon

Correspondence to :
Intan Muchtadi-Alamsyah (ntan@math.itb.ac.id)

Received: December 16, 2020; Revised: July 26, 2021; Accepted: August 9, 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.

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 μ was introduced by Zadeh [1]. This assigns a number from the unit interval [0,1] to each member of the universe of discourse to indicate the degree of belonging to the collection under consideration. The principle of fuzzy sets generalizes the theory of classical sets by facilitating intermediate situations between the whole and nothing. A membership function in a fuzzy set is introduced to characterize the membership degree of an entity in a class. The degree of membership lies between values of 0 and 1, which indicate that the element does not belong and belongs to a class, respectively. The remaining values then indicate the degree to which the class belongs. In fuzzy set theory, the membership function has replaced the characteristic function of the crisp set theory. Since Zadeh’s groundbreaking study [1], the theory of fuzzy sets has been used in various fields such as artificial intelligence, automata theory, engineering, life sciences, mathematics, management sciences, medical sciences, signal processing, social sciences, and statistics [2].

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 [512].

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 [1517] 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 [2529] 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:

Definition 1 [18]

Let be a non-empty set. Then, a picture fuzzy set (PFS) on is defined as

S={(ζ,m(ζ),a(ζ),n(ζ)):ζU},

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 .

Definition 2 [8]

Let and be two PFSs described on , which is the universe of discourse. Then

  • the union of and is

    S1S2={(ζ,mS1(ζ)mS2(ζ),aS1(ζ)aS2(ζ),nS1(ζ)nS2(ζ)):ζU},

  • the intersection of and is

    S1S2={(ζ,mS1(ζ)mS2(ζ),aS1(ζ)aS2(ζ),nS1(ζ)nS2(ζ)):ζU},

  • the symmetric difference of and is

    S1-S2={(ζ,mS1-S2(ζ),aS1-S2(ζ),nS1-S2(ζ)):ζU},

    where

    mS1-S2(ζ)=0mS1-mS2,nS1-S2(ζ)=0nS1-nS2,aS1-S2(ζ)={1-mS1-S2(ζ)-nS1-S2(ζ),if aS1(ζ)>aS2(ζ),{1+aS1(ζ)-aS2(ζ)}{1-mS1-S2(ζ)-nS1-S2(ζ)},if aS1(ζ)aS2(ζ),

  • if and only if

    mS1(ζ)mS2(ζ),   aS1(ζ)aS2(ζ),

    and

    nS1(ζ)nS2(ζ),ζU.

Definition 3

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 .

Definition 4 [31]

A near-ring N is a non-void set, with ‘+’ and ‘.’ as two binary operations with the following constraints:

  • (i) (N, +) is a group;

  • (ii) (N, ·) is a semi-group; and

  • (iii) , for all .

To be precise, because N fulfills the left distributive law, it is a left near-ring. Instead of a “left near-ring,” the term we use is a “near-ring.” Recall that and for all ; however, usually for some .

Definition 5 [31]

A non-empty set of a near-ring N is an ideal of N if the following axioms are satisfied:

  • (i) ( , +) is a normal subgroup of (N, +);

  • (ii) ; and

  • (iii) for all and all .

If satisfies (i) and (ii) of Definition 5, then is a left ideal of N. If satisfies (i) and (iii) of Definition 5, then is the right ideal of N.

Definition 6 [13]

A fuzzy set , where N is a near-ring, is said to be a fuzzy subnear-ring of N if

  • (i) ,

  • (ii) , for all

Definition 7 [32]

A fuzzy set , where N is a near-ring, is called the fuzzy ideal of N, if the following axioms are satisfied:

  • (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 N, and if it satisfies (i), (ii) and (iv), then it is said to be a fuzzy right ideal of N.

Definition 8

A fuzzy set , where N is a near-ring, is said to be an anti-fuzzy ideal of N, if the following axioms are satisfied:

  • (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.

3.1 Picture Fuzzy Ideals of Near-Rings: Definitions and Examples

Definition 9

A picture fuzzy set in a near-ring N is said to be a picture fuzzy subnear-ring of N if

  • (i) ; and

  • (ii) .

Definition 10

Let N be a nearby ring. A picture fuzzy set in a near-ring N is said to be a picture fuzzy ideal of N if the following axioms are satisfied: For all ,

  • (i) ;

  • (ii) ;

  • (iii) ; and

  • (iv) .

A picture fuzzy subset is said to be a picture fuzzy left ideal of N if it satisfies (i)–(iii), and a picture fuzzy subset is said to be a picture fuzzy right ideal of N if it satisfies (i), (ii), and (iv).

Remark 1

Let (N, +, ·) be any near-ring, and be a PFS on N. If is a picture fuzzy ideal of N, it is then a picture fuzzy left ideal of N (a picture fuzzy right ideal of N), and as a result, it is a picture fuzzy subnear-ring of N.

Proposition 1

Let N be a near-ring, and be a PFS of N. If is a picture fuzzy subnear-ring (left ideal or right ideal or ideal) of N, then the following statements are true:

  • (i) for all ; and

  • (ii) for all .

Proof

(1) Let be a picture fuzzy subnear-ring (left ideal, right ideal, or ideal) of N. Then, condition (i) of Definition 9 holds. This implies that , and .

(2) Substituting in Condition (i) of Definition 9, we obtain for all . Because , it follows that , and for all . This completes the proof.

Remark 2

Let (N, +, ·) be any near-ring, a, b, c be fixed values in the unit interval with 0 ≤ a + b + c ≤ 1, and define the PFS on N as follows:

mP(u)=a,aP(u)=b,nP(u)=cfor all uN.

Then, is a picture fuzzy ideal of N.

Example 1

Let N1 = {0, 1, 2, 3} with the two binary operations “+1” and “·1” be defined as follows:

+10123
00123
11230
22301
33012


and

·10123
00000
10123
20123
30123


Then, (N1, +1, ·1) is a near-ring. Let the picture fuzzy set be defined as . It can be clearly seen that the picture fuzzy set is a picture fuzzy ideal of N1.

Example 2

Let ℤ be the set of integers. Then, (ℤ, +, ·) is a near-ring. Here, “+” is a standard addition and “·” is defined as a · b = b for all a, b ∈ ℤ. We define the picture fuzzy set on ℤ as follows: For all x ∈ ℤ,

mP(x)={0.8if xis a multiple of 4;0.2otherwise,aP(x)={0.1if xis a multiple of 4;0.35otherwise,

and

nP(x)={0.05if xis a multiple of 4;0.4otherwise,

PFS on ℤ is as follows: For all x ∈ ℤ,

mP'(x)={0.7if xis a multiple of 3;0.3otherwise,aP'(x)={0.05if xis a multiple of 3;0.36otherwise,

and

nP'(x)={0.1if xis a multiple of 3;0.33otherwise.

It can be clearly seen that the picture fuzzy sets are picture fuzzy ideals of ℤ.

Example 3

Let N2 = {e, f, g, h} with the two binary operations “+2” and “·2” be defined as follows:

+2efgh
eefgh
ffehg
ggfhe
hhgef


and

·2efgh
eeeee
feeee
geeee
heeff


Then, (N2, +2, ·2) is a near-ring. Let the picture fuzzy set be defined as follows: . It can be clearly seen that the picture fuzzy set is a picture fuzzy ideal of N2.

3.2 Operations on Picture Fuzzy Ideals of Near-Rings

In this subsection, we study the different types of operations on picture fuzzy ideals under near-rings.

Theorem 1

If and are any two picture fuzzy ideals of N, then is a picture fuzzy ideal of N.

Proof

Let and be two picture fuzzy ideals of N. Let .

  • (i) .

  • (ii) .

  • (iii) .

  • (iv) .

Therefore, is a picture fuzzy ideal of N.

Corollary 1

If are picture fuzzy ideals of N, then A=i=1nAi is again a picture fuzzy ideal of N.

Proof

The proof follows from Theorem 1 and applies induction.

Example 4

Consider (ℤ, +, ·) as a near-ring, , and let be picture fuzzy sets and their corresponding picture fuzzy ideals presented in Example 2. Then, the picture fuzzy set on ℤ is defined as follows: For all x ∈ ℤ,

mPP'(x)={0.7,if xis a multiple of 12;0.3,if xis a multiple of 4but is not a multiple of 3;0.2,otherwise,aPP'(x)={0.1,if xis a multiple of 12;0.35,if xis a multiple of 3but is not a multiple of 4;0.36,otherwise,

and

nPP'(x)={0.1,if xis a multiple of 12;0.33,if xis a multiple of 4but is not a multiple of 3;0.4,otherwise.

Using Theorem 1, we find that the picture fuzzy set is a picture fuzzy ideal of ℤ.

Remark 3

The union of the picture fuzzy ideals of a near-ring N is not necessarily a picture fuzzy ideal of N.

We illustrate Remark 3 through the following example.

Example 5

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 .

Definition 11

Let and be two picture fuzzy subsets of two near-rings N1 and N2, respectively. Then, the direct product of the picture fuzzy subsets of the near-rings is defined as , such that and , where .

Similarly, we can define the direct product of n picture fuzzy subsets of the near-rings Ni with i = 1, . . ., n.

Remark 4

Let and be two picture fuzzy subsets of two near-rings, N1 and N2, respectively. Then, is a picture fuzzy ideal of N1 × N2 if it satisfies the following conditions:

  • (i) .

  • (ii) .

  • (iii) .

  • (iv) .

Theorem 2

Let and be two picture fuzzy ideals of two near-rings N1 and N2, respectively. Then, is a picture fuzzy ideal of N1 × N2.

Proof

Let and be two picture fuzzy ideals of N1 and N2, respectively.

Let .

  • (i) .

  • (ii) .

  • (iii) .

  • (iv) .

Corollary 2

Let n be a positive integer and be a picture fuzzy ideal of the near-rings Ni for i = 1, . . ., n. Then, i=1nAi is a picture fuzzy ideal of i=1nNi.

Proof

The proof follows from Theorem 2 and through induction.

Example 6

Let (N1, +1, ·1), , and (ℤ, +, ·), be the near-rings with their corresponding PFS presented in Examples 1 and 2, respectively. By defining the picture fuzzy set on N1 × ℤ as follows:

mP1×P((x,y))={0.8if x=0and yis a multiple of 4;0.4if x0and yis a multiple of 4;0.2otherwise,aP1×P((x,y))={0.15if x=0and yis a multiple of 4;0.2if x0and yis a multiple of 4;0.35otherwise,

and

nP1×P((x),y)={0.05if x=0and yis a multiple of 4;0.4if yis not a multiple of 4;0.3if x0and yis a multiple of 4,

we obtain through Theorem 2 that is a picture fuzzy ideal of N1 × ℤ.

3.3 Homomorphism of Near-Rings through Picture Fuzzy Ideals

We study picture fuzzy ideals under near-ring homomorphisms.

Definition 12

A map , where N1 and N2 are near-rings, is said to be a homomorphism if and , for all .

Let be a map, where N1 and N2 are near-rings. For any picture fuzzy set in N2, we define the picture fuzzy set Ag=(mAg,aAg,nAg) in N1 by mAg(u)=mU(g(u)),aAg(u)=aA(g(u)),nAg(u)=nA(g(u)). for all .

Theorem 3

Let be a homomorphism, where N1 and N2 are near-rings. If the picture fuzzy set in N2 is a picture fuzzy ideal of N2, then the picture fuzzy set Ag=(mAg,aAg,nAg) in N1 is a picture fuzzy ideal of N1.

Proof

Because is a homomorphism, it follows that for any , and , we obtain the following.

  • (i) mAg(u-v)=mA(g(u)-g(v))mA(g(u))mA(g(v))=mAg(u)mAg(v),aAg(u-v)=aA(g(u)-g(v))aA(g(u))aA(g(v))=aAg(u)aAg(v),nAg(u-v)=nA(g(u)-g(v))nA(g(u))nA(g(v))=nAg(u)nAg(v)..

  • (ii) mAg(v+u-v)=mA(g(v)+g(u)-g(v))mA(g(u))=mAg(u),aAg(v+u-v)=aA(g(v)+g(u)-g(v))aA(g(u))=aAg(u),nAg(v+u-v)=nA(g(v)+g(u)-g(v))nA(g(u))=nAg(u)..

  • (iii) mAg(uv)=mA(g(u)g(v))mA(g(v))=mAg(v),aAg(uv)=aA(g(u)g(v))aA(g(v))=aAg(v),nAg(uv)=nA(g(u)g(v))nA(g(v))=nAg(v)..

  • (iv) mAg((u+m)v-uv)=mA(g(u+m)v-uv))=mA((g(u)+g(m))g(v)-g(u)g(v))mA(g(m))=mAg(m),aAg((u+m)v-uv)=aA(g((u+m)v-uv))=aA((g(u)+g(m))g(v)-g(u)g(v))aA(g(m))=aAg(m),nAg((u+m)v-uv)=nA(g((u+m)v-uv))=nA((g(u)+g(m))g(v)-g(u)g(v))nA(g(m))=nAg(m)..

Example 7

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 a · b = b for all a, b ∈ 2ℤ, g: 2ℤ → ℤ is the near-ring homomorphism defined through the inclusion map. Theorem 3 asserts that the picture fuzzy set Pg=(mPg,aPg,nPg) in 2ℤ is a picture fuzzy ideal of 2ℤ.

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.

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Article

Original Article

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.

On Picture Fuzzy Ideals of Near-Rings

Mani Parimala1, Intan Muchtadi-Alamsyah2, and Madeleine Al-Tahan3

1Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamilnadu, India
2AlgebraResearch Division and Center of Artificial Intelligence, Institut Teknologi Bandung, Bandung, Indonesia
3Department of Mathematics, Lebanese International University, Beirut, Lebanon

Correspondence to:Intan Muchtadi-Alamsyah (ntan@math.itb.ac.id)

Received: December 16, 2020; Revised: July 26, 2021; Accepted: August 9, 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

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

1. Introduction

The definition of a fuzzy set with a membership function μ was introduced by Zadeh [1]. This assigns a number from the unit interval [0,1] to each member of the universe of discourse to indicate the degree of belonging to the collection under consideration. The principle of fuzzy sets generalizes the theory of classical sets by facilitating intermediate situations between the whole and nothing. A membership function in a fuzzy set is introduced to characterize the membership degree of an entity in a class. The degree of membership lies between values of 0 and 1, which indicate that the element does not belong and belongs to a class, respectively. The remaining values then indicate the degree to which the class belongs. In fuzzy set theory, the membership function has replaced the characteristic function of the crisp set theory. Since Zadeh’s groundbreaking study [1], the theory of fuzzy sets has been used in various fields such as artificial intelligence, automata theory, engineering, life sciences, mathematics, management sciences, medical sciences, signal processing, social sciences, and statistics [2].

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 [512].

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 [1517] 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 [2529] 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.

2. Preliminaries

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:

Definition 1 [18]

Let be a non-empty set. Then, a picture fuzzy set (PFS) on is defined as

S={(ζ,m(ζ),a(ζ),n(ζ)):ζU},

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 .

Definition 2 [8]

Let and be two PFSs described on , which is the universe of discourse. Then

  • the union of and is

    S1S2={(ζ,mS1(ζ)mS2(ζ),aS1(ζ)aS2(ζ),nS1(ζ)nS2(ζ)):ζU},

  • the intersection of and is

    S1S2={(ζ,mS1(ζ)mS2(ζ),aS1(ζ)aS2(ζ),nS1(ζ)nS2(ζ)):ζU},

  • the symmetric difference of and is

    S1-S2={(ζ,mS1-S2(ζ),aS1-S2(ζ),nS1-S2(ζ)):ζU},

    where

    mS1-S2(ζ)=0mS1-mS2,nS1-S2(ζ)=0nS1-nS2,aS1-S2(ζ)={1-mS1-S2(ζ)-nS1-S2(ζ),if aS1(ζ)>aS2(ζ),{1+aS1(ζ)-aS2(ζ)}{1-mS1-S2(ζ)-nS1-S2(ζ)},if aS1(ζ)aS2(ζ),

  • if and only if

    mS1(ζ)mS2(ζ),   aS1(ζ)aS2(ζ),

    and

    nS1(ζ)nS2(ζ),ζU.

Definition 3

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 .

Definition 4 [31]

A near-ring N is a non-void set, with ‘+’ and ‘.’ as two binary operations with the following constraints:

  • (i) (N, +) is a group;

  • (ii) (N, ·) is a semi-group; and

  • (iii) , for all .

To be precise, because N fulfills the left distributive law, it is a left near-ring. Instead of a “left near-ring,” the term we use is a “near-ring.” Recall that and for all ; however, usually for some .

Definition 5 [31]

A non-empty set of a near-ring N is an ideal of N if the following axioms are satisfied:

  • (i) ( , +) is a normal subgroup of (N, +);

  • (ii) ; and

  • (iii) for all and all .

If satisfies (i) and (ii) of Definition 5, then is a left ideal of N. If satisfies (i) and (iii) of Definition 5, then is the right ideal of N.

Definition 6 [13]

A fuzzy set , where N is a near-ring, is said to be a fuzzy subnear-ring of N if

  • (i) ,

  • (ii) , for all

Definition 7 [32]

A fuzzy set , where N is a near-ring, is called the fuzzy ideal of N, if the following axioms are satisfied:

  • (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 N, and if it satisfies (i), (ii) and (iv), then it is said to be a fuzzy right ideal of N.

Definition 8

A fuzzy set , where N is a near-ring, is said to be an anti-fuzzy ideal of N, if the following axioms are satisfied:

  • (i) ;

  • (ii) ;

  • (iii) ;

  • (iv) for all .

3. Picture Fuzzy Subsets of Near-Rings

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.

3.1 Picture Fuzzy Ideals of Near-Rings: Definitions and Examples

Definition 9

A picture fuzzy set in a near-ring N is said to be a picture fuzzy subnear-ring of N if

  • (i) ; and

  • (ii) .

Definition 10

Let N be a nearby ring. A picture fuzzy set in a near-ring N is said to be a picture fuzzy ideal of N if the following axioms are satisfied: For all ,

  • (i) ;

  • (ii) ;

  • (iii) ; and

  • (iv) .

A picture fuzzy subset is said to be a picture fuzzy left ideal of N if it satisfies (i)–(iii), and a picture fuzzy subset is said to be a picture fuzzy right ideal of N if it satisfies (i), (ii), and (iv).

Remark 1

Let (N, +, ·) be any near-ring, and be a PFS on N. If is a picture fuzzy ideal of N, it is then a picture fuzzy left ideal of N (a picture fuzzy right ideal of N), and as a result, it is a picture fuzzy subnear-ring of N.

Proposition 1

Let N be a near-ring, and be a PFS of N. If is a picture fuzzy subnear-ring (left ideal or right ideal or ideal) of N, then the following statements are true:

  • (i) for all ; and

  • (ii) for all .

Proof

(1) Let be a picture fuzzy subnear-ring (left ideal, right ideal, or ideal) of N. Then, condition (i) of Definition 9 holds. This implies that , and .

(2) Substituting in Condition (i) of Definition 9, we obtain for all . Because , it follows that , and for all . This completes the proof.

Remark 2

Let (N, +, ·) be any near-ring, a, b, c be fixed values in the unit interval with 0 ≤ a + b + c ≤ 1, and define the PFS on N as follows:

mP(u)=a,aP(u)=b,nP(u)=cfor all uN.

Then, is a picture fuzzy ideal of N.

Example 1

Let N1 = {0, 1, 2, 3} with the two binary operations “+1” and “·1” be defined as follows:

+10123
00123
11230
22301
33012


and

·10123
00000
10123
20123
30123


Then, (N1, +1, ·1) is a near-ring. Let the picture fuzzy set be defined as . It can be clearly seen that the picture fuzzy set is a picture fuzzy ideal of N1.

Example 2

Let ℤ be the set of integers. Then, (ℤ, +, ·) is a near-ring. Here, “+” is a standard addition and “·” is defined as a · b = b for all a, b ∈ ℤ. We define the picture fuzzy set on ℤ as follows: For all x ∈ ℤ,

mP(x)={0.8if xis a multiple of 4;0.2otherwise,aP(x)={0.1if xis a multiple of 4;0.35otherwise,

and

nP(x)={0.05if xis a multiple of 4;0.4otherwise,

PFS on ℤ is as follows: For all x ∈ ℤ,

mP'(x)={0.7if xis a multiple of 3;0.3otherwise,aP'(x)={0.05if xis a multiple of 3;0.36otherwise,

and

nP'(x)={0.1if xis a multiple of 3;0.33otherwise.

It can be clearly seen that the picture fuzzy sets are picture fuzzy ideals of ℤ.

Example 3

Let N2 = {e, f, g, h} with the two binary operations “+2” and “·2” be defined as follows:

+2efgh
eefgh
ffehg
ggfhe
hhgef


and

·2efgh
eeeee
feeee
geeee
heeff


Then, (N2, +2, ·2) is a near-ring. Let the picture fuzzy set be defined as follows: . It can be clearly seen that the picture fuzzy set is a picture fuzzy ideal of N2.

3.2 Operations on Picture Fuzzy Ideals of Near-Rings

In this subsection, we study the different types of operations on picture fuzzy ideals under near-rings.

Theorem 1

If and are any two picture fuzzy ideals of N, then is a picture fuzzy ideal of N.

Proof

Let and be two picture fuzzy ideals of N. Let .

  • (i) .

  • (ii) .

  • (iii) .

  • (iv) .

Therefore, is a picture fuzzy ideal of N.

Corollary 1

If are picture fuzzy ideals of N, then A=i=1nAi is again a picture fuzzy ideal of N.

Proof

The proof follows from Theorem 1 and applies induction.

Example 4

Consider (ℤ, +, ·) as a near-ring, , and let be picture fuzzy sets and their corresponding picture fuzzy ideals presented in Example 2. Then, the picture fuzzy set on ℤ is defined as follows: For all x ∈ ℤ,

mPP'(x)={0.7,if xis a multiple of 12;0.3,if xis a multiple of 4but is not a multiple of 3;0.2,otherwise,aPP'(x)={0.1,if xis a multiple of 12;0.35,if xis a multiple of 3but is not a multiple of 4;0.36,otherwise,

and

nPP'(x)={0.1,if xis a multiple of 12;0.33,if xis a multiple of 4but is not a multiple of 3;0.4,otherwise.

Using Theorem 1, we find that the picture fuzzy set is a picture fuzzy ideal of ℤ.

Remark 3

The union of the picture fuzzy ideals of a near-ring N is not necessarily a picture fuzzy ideal of N.

We illustrate Remark 3 through the following example.

Example 5

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 .

Definition 11

Let and be two picture fuzzy subsets of two near-rings N1 and N2, respectively. Then, the direct product of the picture fuzzy subsets of the near-rings is defined as , such that and , where .

Similarly, we can define the direct product of n picture fuzzy subsets of the near-rings Ni with i = 1, . . ., n.

Remark 4

Let and be two picture fuzzy subsets of two near-rings, N1 and N2, respectively. Then, is a picture fuzzy ideal of N1 × N2 if it satisfies the following conditions:

  • (i) .

  • (ii) .

  • (iii) .

  • (iv) .

Theorem 2

Let and be two picture fuzzy ideals of two near-rings N1 and N2, respectively. Then, is a picture fuzzy ideal of N1 × N2.

Proof

Let and be two picture fuzzy ideals of N1 and N2, respectively.

Let .

  • (i) .

  • (ii) .

  • (iii) .

  • (iv) .

Corollary 2

Let n be a positive integer and be a picture fuzzy ideal of the near-rings Ni for i = 1, . . ., n. Then, i=1nAi is a picture fuzzy ideal of i=1nNi.

Proof

The proof follows from Theorem 2 and through induction.

Example 6

Let (N1, +1, ·1), , and (ℤ, +, ·), be the near-rings with their corresponding PFS presented in Examples 1 and 2, respectively. By defining the picture fuzzy set on N1 × ℤ as follows:

mP1×P((x,y))={0.8if x=0and yis a multiple of 4;0.4if x0and yis a multiple of 4;0.2otherwise,aP1×P((x,y))={0.15if x=0and yis a multiple of 4;0.2if x0and yis a multiple of 4;0.35otherwise,

and

nP1×P((x),y)={0.05if x=0and yis a multiple of 4;0.4if yis not a multiple of 4;0.3if x0and yis a multiple of 4,

we obtain through Theorem 2 that is a picture fuzzy ideal of N1 × ℤ.

3.3 Homomorphism of Near-Rings through Picture Fuzzy Ideals

We study picture fuzzy ideals under near-ring homomorphisms.

Definition 12

A map , where N1 and N2 are near-rings, is said to be a homomorphism if and , for all .

Let be a map, where N1 and N2 are near-rings. For any picture fuzzy set in N2, we define the picture fuzzy set Ag=(mAg,aAg,nAg) in N1 by mAg(u)=mU(g(u)),aAg(u)=aA(g(u)),nAg(u)=nA(g(u)). for all .

Theorem 3

Let be a homomorphism, where N1 and N2 are near-rings. If the picture fuzzy set in N2 is a picture fuzzy ideal of N2, then the picture fuzzy set Ag=(mAg,aAg,nAg) in N1 is a picture fuzzy ideal of N1.

Proof

Because is a homomorphism, it follows that for any , and , we obtain the following.

  • (i) mAg(u-v)=mA(g(u)-g(v))mA(g(u))mA(g(v))=mAg(u)mAg(v),aAg(u-v)=aA(g(u)-g(v))aA(g(u))aA(g(v))=aAg(u)aAg(v),nAg(u-v)=nA(g(u)-g(v))nA(g(u))nA(g(v))=nAg(u)nAg(v)..

  • (ii) mAg(v+u-v)=mA(g(v)+g(u)-g(v))mA(g(u))=mAg(u),aAg(v+u-v)=aA(g(v)+g(u)-g(v))aA(g(u))=aAg(u),nAg(v+u-v)=nA(g(v)+g(u)-g(v))nA(g(u))=nAg(u)..

  • (iii) mAg(uv)=mA(g(u)g(v))mA(g(v))=mAg(v),aAg(uv)=aA(g(u)g(v))aA(g(v))=aAg(v),nAg(uv)=nA(g(u)g(v))nA(g(v))=nAg(v)..

  • (iv) mAg((u+m)v-uv)=mA(g(u+m)v-uv))=mA((g(u)+g(m))g(v)-g(u)g(v))mA(g(m))=mAg(m),aAg((u+m)v-uv)=aA(g((u+m)v-uv))=aA((g(u)+g(m))g(v)-g(u)g(v))aA(g(m))=aAg(m),nAg((u+m)v-uv)=nA(g((u+m)v-uv))=nA((g(u)+g(m))g(v)-g(u)g(v))nA(g(m))=nAg(m)..

Example 7

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 a · b = b for all a, b ∈ 2ℤ, g: 2ℤ → ℤ is the near-ring homomorphism defined through the inclusion map. Theorem 3 asserts that the picture fuzzy set Pg=(mPg,aPg,nPg) in 2ℤ is a picture fuzzy ideal of 2ℤ.

4. Conclusion

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.

+10123
00123
11230
22301
33012

·10123
00000
10123
20123
30123

+2efgh
eefgh
ffehg
ggfhe
hhgef

·2efgh
eeeee
feeee
geeee
heeff

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