# Picture Fuzzy Ideals of Near-Rings

Authors: Awais Asif; Hassen Aydi; Muhammad Arshad; Abdul Rehman; Usman Tariq
Journal: Journal of Mathematics
 (2020)

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## Abstract

A picture fuzzy set (PFS) is an augmentation of Atanassov’s intuitionistic fuzzy set (IFS). The PFS-based models are useful in the circumstances when we face uncertain and vague information, especially in the case when we need more answers of the form “indeed,” “avoid,” “no,” and “refusal.” It has been considered as an essential tool to deal with unsure data during an investigation. In this manuscript, we explore the idea of a picture fuzzy near-ring (PFNR) and a picture fuzzy ideal (PFI) of a near-ring (NR). We illustrate some basic properties such as union, intersection, homomorphic image, and preimage of PFIs of a NR. Furthermore, there is discussion about the direct product of PFIs of a NR.


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## Body

## 1. Introduction

Fuzzy set (FS) was first introduced by Zadeh [1]. It discusses the grade of truth values belonging to a unit interval. FS is a valuable technique to cope with vague and difficult information in a real-life decision. It has received extensive attention from researchers and has been utilized in decision-making in many ways (see [2, 3]). However, in some cases, FS cannot work reliably. For coping such types of issues, Atanassov [4] explored the intuitionistic fuzzy set (IFS) to deal with more complex problems. IFS contains truth and falsity grades, whose sum is belonging to a unit interval. IFS is a more modified version of a FS, and many researchers have used it in decision-making [5], such as medical diagnosis [6] and pattern recognition [7]. Although FS and IFS are applicable in various fields, however, there exist problems when a decision maker face various kinds of opinions of human beings such as “yes,” “abstinence,” “no,” and “refusal.” To coping with such kinds of problems, Cuong [8] explored the concept of a picture fuzzy set (PFS), containing the grades of truth, abstinence, falsity, and refusal, whose sum is belonging to a unit interval. To deal with uncertain and difficult information in real-life decisions, PFS is more effective than the IFS and FS. Joshi [9] presented a new setting as criteria of fuzzy entropy for the PFS and proposed a picture fuzzy information model based on Tsallis–Havrda–Charvat entropy. Qyias et al. [10] established aggregation operators based on linguistic PFSs and their application in decision-making problems, while Ganie et al. [11] examined the new correlation coefficient for the PFS.The concept of an ideal in the frame of a near-ring is discussed by numerous scholars by using intuitionistic fuzzy sets. To investigate complicated and awkward information, a picture fuzzy set is a more influential tool than the fuzzy set and intuitionistic fuzzy sets. Thus, the concept of a picture fuzzy ideal of near-rings is presented in this article. Our results clearly show that the idea of a picture fuzzy set is more proficient and reliable than the aforementioned concepts. The notion of fuzzy sub-near-rings (FSNR) and fuzzy ideals (FI) of a NR was introduced by Abou-Zaid [12]. This concept was further discussed by Kim [13] and Dutta and Biswas [14]. Chinnadurai and Kadalarasi [15] discussed the direct product of n (n = 1, 2, …, k) FSNR, FI, and fuzzy R-subgroups. The advantages of a PFS became the motivation of our main theorems. The results presented are entirely new and more beneficial than the existing results in the literature. The aims of this manuscript are given as follows:(1)
To explore the idea of a picture fuzzy near-ring (PFNR) and a picture fuzzy ideal (PFI) of a near-ring (NR)(2)
To describe some basic properties such as union, intersection, homomorphic image, and preimage of PFIs of a NR(3)
To elaborate the direct product of PFIs of a NRThe summary of this manuscript is as follows: in Section2, we review some notions such as groups, subgroups, ideals, and fuzzy sets. In Section 3, we explore the idea of a picture fuzzy near-ring (PFNR) and a picture fuzzy ideal (PFI) of a near-ring (NR). In Section 4, we investigate some basic properties such as union, intersection, homomorphic image, and preimage of a PFI of a NR. In Section 5, we discuss about the direct product of PFIs of a NR. The conclusion of this manuscript is discussed in Section 6.


## 2. Preliminaries

The aim of this study is to recall some basic notions such as near-rings, ideals of a near-ring, fuzzy sub-near-rings, fuzzy ideals, and picture fuzzy sets.Definition 1 (see [13]).
For any nonempty setE with two binary operations “+” and “⋅,” we say that E is called a near-ring if the following conditions hold:(i)
E,+ is a group(ii)
E,⋅ is a semigroup(iii)
h+k⋅t=h⋅t+k⋅t for h,k,t∈EDefinition 2 (see [14]).
An ideal of a NRE is a subgroup A of E such that(i)
A,+ is a normal subgroup of E,+,(ii)
AE⊆A,(iii)
kh+i−kh∈A, for h,k∈E and i∈A,whereA is a left ideal of E whenever A satisfies (i) and (ii) and a right ideal of E if A satisfies (i) and (iii).Definition 3 (see [16]).
A FS in a NRE is said to be a fuzzy sub-near-ring of E if(i)
Bh−k≥minBh,Bk, for all h,k∈E(ii)
Bhk≥minBh,Bk, for all h,k∈EDefinition 4 (see [16]).
A FSB in a NR E is said to be a fuzzy ideal of E if(i)
Bh−k≥minBh,Bk, for all h,k∈E(ii)
Bk+h−k≥Bh, for all h,k∈E(iii)
Bhk≥Bk, for all h,k∈E(iv)
Bh+tk−hk≥Bt, for all h,k∈E
A fuzzy set satisfying (i)–(iii) is called a fuzzy left ideal ofE, whereas a fuzzy set with (i) and (ii) is called a fuzzy right ideal of E.Definition 5 (see [8]).
A PFSP on the universe of discourse U characterized by a truth membership function BPh, an indeterminacy function APh, and a falsity membership function FPh is defined as(1)P=h,BPh,APh,FPh:h∈U,where BP,AP,FP:U⟶0,1 and 0≤BPh+APh+FPh≤1.Definition 6 (see [8]).
LetP and Q be PFSs of U. Then,(2)P∪Q=h,BP∪Qh,AP∪Qh,FP∪Qh,where(3)BP∪Qh=maxBPh,BQh,AP∪Qh=minAPh,AQh,FP∪Qh=minFPh,FQh,for all h∈U.(4)P∩Q=h,BP∩Qh,AP∩Qh,FP∩Qh,where(5)BP∩Qh=minBPh,BQh,AP∩Qh=maxAPh,AQh,FP∩Qh=maxFPh,FQh,for all h∈U.


## 3. Picture Fuzzy Ideals of Near-Rings

The aim of this study is to explore the idea of a PFNR and a PFI of a NR.Definition 7.
A PFSP in a NR E is called a PFSNR of E if(i)
BPh−k≥minBPh,BPk,APh−k≤maxAPh,APk,FPh−k≤maxFPh,FPk.(ii)
BPhk≥minBPh,BPk,APhk≤maxAPh,APkandFPhk≤maxFPh,FPk.Definition 8.
LetE be a NR. A PFS P in a NR E is called a PFS of E if(i)
BPh−k≥minBPh,BPk,APh−k≤maxAPh,APk,andFPh−k≤maxFPh,FPh.(ii)
BPk+h−k≥BPh,APk+h−k≤APhandFPk+h−k≤FPh.(iii)
BPhk≥BPk,APhk≤APk,andFPhk≤FPk.(iv)
BPh+tk−hk≥BPt,APh+tk−hk≤APt,andFPh+tk−hk≤FPt.
A picture fuzzy subset with the above conditions (i)–(iii) is called a picture fuzzy left ideal ofE, whereas a picture fuzzy subset with (i), (ii), and (iv) is called a picture fuzzy right ideal of E.Theorem 1.
LetP and Q be PFIs of E. If P⊂Q, then P∪Q is a PFI of E.Proof.
LetP and Q be PFIs of E. Let h,k,t∈E; then,(6)BP∪Qh−k=maxBPh−k,BQh−k≥maxminBPh,BPk,minBQh,BQk=minmaxBPh,BQh,maxBPk,BQk=minBP∪Qh,BP∪Qk.
Similarly, for the abstinence degree, we write(7)AP∪Qh−k=minAPh−k,AQh−k≤minmaxAPh,APk,maxAQh,AQk=maxminAPh,AQh,maxAPk,AQk=maxAP∪Qh,AP∪Qk,and for falsity grade, we have(8)FP∪Qh−k=minFPh−k,FQh−k,≤minmaxFPh,FPk,maxFQh,FQk=maxminFPh,FQh,maxFPk,FQk=maxFP∪Qh,FP∪Qk.
Next, we write(9)BP∪Qk+h−k=maxBPk+h−k,BQk+h−k,≥maxBPh,BQh,=BP∪Qh.
For the abstinence degree, we obtain(10)AP∪Qk+h−k=minAPk+h−k,AQk+h−k,≤minAPh,AQh=AP∪Qh,and for falsity grade, we get(11)FP∪Qk+h−k=minFPk+h−k,FQk+h−k,≤minFPh,FQh,=FP∪Qh.
Furthermore, we deduce that(12)BP∪Qhk=maxBPhk,BQhk≥maxBPk,BQk=BP∪Qk,AP∪Qhk=minAPhk,AQhk≤minAPk,AQk=AP∪Qk,FP∪Qhk=minFPhk,FQhk≤minFPk,FQk=FP∪Qk.
At last, we obtain(13)BP∪Qh+tk−hk=maxBPh+tk−hk,BQh+tk−hk≥maxBPt,BQt=BP∪Qt,AP∪Qh+tk−hk=minAPh+tk−hk,AQh+tk−hk≤minAPt,AQt=AP∪Qt,BP∪Qh+tk−hk=minFPh+tk−hk,FQh+tk−hk≤minFPt,FQt=FP∪Qt.
Therefore,P∪Q is a PFI of E.Theorem 2.
LetP and Q be PFIs of E. If P⊂Q, then P∩Q is a PFI of E.Proof.
LetP and Q be PFIs of E. Let h,k,t∈E. Then, the following are obtained.
For truth grade, we get(14)BP∩Qh−k=minBPh−k,BQh−k≥minminBPh,BPk,minBQh,BQk=minminBPh,BQh,minBPk,BQk=minBP∩Qh,BP∩Qk.
Similarly, for abstinence grade, we write(15)AP∩Qh−k=maxAPh−k,AQh−k≤maxmaxAPh,APk,maxAQh,AQk=maxmaxAPh,AQh,maxAPk,AQk=maxAP∩Qh,AP∩Qk,and for falsity grade, we obtain(16)FP∩Qh−k=maxFPh−k,FQh−k≤maxmaxFPh,FPk,maxFQh,FQk=maxmaxFPh,FQh,maxFPk,FQk,=maxFP∩Qh,FP∩Qk.
Next, we obtain(17)BP∩Qk+h−k=minBPk+h−k,BQk+h−k≥minBPh,BQh=BP∩Qh.
Similarly,(18)AP∩Qk+h−k=maxAPk+h−k,AQk+h−k≤maxAPh,AQh=AP∩Qh,FP∩Qk+h−k=maxFPk+h−k,FQk+h−k≤maxFPh,FQh=FP∩Qh.
Furthermore, we deduce that(19)BP∩Qhk=minBPhk,BQhk≥minBPk,BQk=BP∩Qk,AP∩Qhk=maxAPhk,AQhk≤maxAPk,AQk=AP∩Qk,FP∩Qhk=maxFPhk,FQhk≤maxFPk,FQk=FP∩Qk.
Finally, we conclude that(20)BP∩Qh+tk−hk=minBPh+tk−hk,BQh+tk−hk≥minBPt,BQt=BP∩Qt,AP∩Qh+tk−hk=maxAPh+tk−hk,AQh+tk−hk≤maxAPt,AQt=AP∩Qt,BP∩Qh+tk−hk=maxFPh+tk−hk,FQh+tk−hk≤maxFPt,FQt=FP∩Qt.
Therefore,P∩Q is a PFI of E.Corollary 1.
IfP1,P2,…,Pn are PFIs of E, then P=∩i=1nPi is a PFI of E.Lemma 1.
For allh,b∈A and i is any positive integer, if h=b, then(1)
hi≤bi.(2)
minh,bi=minhi,bi.(3)
maxh,bi=maxhi,bi.Theorem 3.
LetP be a PFI of E. Then,(21)Pm=h,BPmh,APmh,FPmh:h∈Eis a PFI of E, where m is a positive integer and BPmh=BPhm,APmh=APhm, and FPmh=FPhm.Proof.
LetP be a PFI of E. Let h,k,t∈E. Then, the following are observed.
For truth grade, we can write(22)BPmh−k=BPh−km≥minBPh,BPkm=minBPhm,BPkm=minBPmh,BPmk.
Similarly, for abstinence grade, we get(23)APmh−k=APh−km≤maxAPh,APkm=maxAPhm,APkm=maxAPmh,APmk,and for falsity grade, we obtain the following:(24)FPmh−k=FPh−km≤maxFPh,FPkm=maxFPhm,FPkm=maxFPmh,FPmk.
Next, it is obtained that(25)BPmk+h−k=BPk+h−km≥BPhm=BPmh,APmk+h−k=APk+h−km≤APhm=APmh,FPmk+h−k=FPk+h−km≤FPhm=FPmh.
Also, we examine that(26)BPmhk=BPhkm≥BPkm=BPmk,APmhk=APhkm≤APkm=APmk,FPmhk=FPhkm≤FPkm=FPmk.
At last, we write that(27)BPmh+tk−hk=BPh+tk−hkm≥BPtm=BPmt,APmh+tk−hk=APh+tk−hkm≤APtm=APmt,FPmh+tk−hk=FPh+tk−hkm≤FPtm=FPmt.
Therefore,Pm is a PFI of Em.


## 4. Direct Product of Picture Fuzzy Ideals of Near-Rings

This section throws light on some basic properties such as union, intersection, homomorphic image, and preimage of PFIs of a NR.Definition 9.
LetP and Q be picture fuzzy subsets of near-rings (PFSSNR) E1 and E2, respectively. Then, the direct product of PFSSNRs is defined by P×Q:E1×E2⟶0,1 such that(28)P×Q=h,k,BP×Qh,k,AP×Qh,k,FP×Qh,k:h∈E1,k∈E2,where(29)BP×Qh,k=minBPh,BQk,AP×Qh,k=maxAPh,AQk,FP×Qh,k=maxFPh,FQk.Example 1.
SupposeR=a,b,c,d is a near-ring with binary operations “+” and “⋅,” defined by  and , respectively. Suppose PFSs P=αP,βP,γP and Q=αQ,βQ,γQ defined by(30)αPa=αQa=0.7,αPb=αQb=0.5,αPc=αPd=αQc=αQd=0.3,βPa=βQa=0.2,βPb=βQb=0.3,βPc=βPd=βQc=βQd=0.6,γPa=γQa=0.02,γPb=γQb=0.03,γPc=γPd=γQc=γQd=0.07.
Then, we say thatP=αP,βP,γP and Q=αQ,βQ,γQ are PFI of R. Now,(31)P×Q=0.7,0.2,0.02,0.5,0.3,0.03,0.3,0.6,0.06.
Note that this example can be extended to construct an example for Definition10.Definition 10.
LetP and Q be PFSSNRs E1 and E2, respectively. Then, P×Q is a PFI of E1×E2 if it satisfies the following conditions:(32)BP×Qh1,h2−k1,k2≥minBP×Qh1,h2,BP×Qk1,k2,AP×Qh1,h2−k1,k2≤maxAP×Qh1,h2,AP×Qk1,k2,FP×Qh1,h2−k1,k2≤maxFP×Qh1,h2,FP×Qk1,k2,BP×Qk1,k2+h1,h2−k1,k2≥minBP×Qh1,h2,AP×Qk1,k2+h1,h2−k1,k2≤maxAP×Qh1,h2,FP×Qk1,k2+h1,h2−k1,k2≤maxFP×Qh1,h2,BP×Qh1,h2k1,k2≥BP×Qk1,k2,AP×Qh1,h2k1,k2≤AP×Qk1,k2,FP×Qh1,h2k1,k2≤FP×Qk1,k2,BP×Qh1,h2+t1,t2k1,k2−h1,h2k1,k2≥BP×Qt1,t2,AP×Qh1,h2+t1,t2k1,k2−h1,h2k1,k2≤AP×Qt1,t2,FP×Qh1,h2+t1,t2k1,k2−h1,h2k1,k2≤FP×Qt1,t2.Theorem 4.
LetP and Q be PFIs of E1andE2, respectively. Then, P×Q is a PFI of E1×E2.Proof.
LetP and Q be PFIs of E1 andE2, respectively. Let h1,h2,k1,k2,t1,t2∈E1×E2. Then, the following are obtained.
For truth grade, we obtain the following:(33)BP×Qh1,h2−k1,k2=BP×Qh1−k1,h2−k2,=minBPh1−k1,BQh2−k2,≥minminBPh1,BPk1,minBQh2,BQk2,=minminBPh1,BQh1,minBPk1,BQk2,=minBP×Qh1,h2,BP×Qk1,k2.
Similarly, for abstinence grade, we write(34)AP×Qh1,h2−k1,k2=AP×Qh1−k1,h2−k2,=maxAPh1−k1,AQh2−k2,≤maxmaxAPh1,APk1,maxAQh2,AQk2,=maxmaxAPh1,AQh2,maxAPk1,AQk2,=maxAP×Qh1,h2,AP×Qk1,k2,and for falsity grade, we get(35)FP×Qh1,h2−k1,k2=FP×Qh1−k1,h2−k2,=maxFPh1−k1,FQh2−k2,≤maxmaxFPh1,FPk1,maxFQh2,FQk2,=maxmaxFPh1,FQh2,maxFPk1,FQk2,=maxFP×Qh1,h2,FP×Qk1,k2.
Next, it is clear that(36)BP×Qk1,k2+h1,h2−k1,k2=BP×Qk1+h1−k1,k2+h2−k2,=minBPk1+h1−k1,BQk2+h2−k2≥minBPh1,BQh2,=BP×Qh1,h2.
Similarly,(37)AP×Qk1,k2+h1,h2−k1,k2=AP×Qk1+h1−k1,k2+h2−k2,=maxAPk1+h1−k1,AQk2+h2−k2≤maxAPh1,AQh2,=AP×Qh1,h2,FP×Qk1,k2+h1,h2−k1,k2=FP×Qk1+h1−k1,k2+h2−k2,=maxFPk1+h1−k1,FQk2+h2−k2≤maxFPh1,FQh2,=FP×Qh1,h2.
Moreover, we deduce the following inequalities:(38)BP×Qh1,h2k1,k2=BP×Qh1k1,h2k2=minBPh1k1,BQh2k2≥minBPk1,BQk2=BP×Qk1,k2.
Similarly,(39)AP×Qh1,h2k1,k2=AP×Qh1k1,h2k2=maxAPh1k1,AQh2k2≤maxAPk1,AQk2=AP×Qk1,k2,FP×Qh1,h2k1,k2=FP×Qh1k1,h2k2=maxFPh1k1,FQh2k2≤maxFPk1,FQk2=FP×Qk1,k2.
Finally, we prove that(40)BP×Qh1,h2+t1,t2−k1,k2−h1,h2k1,k2=BP×Qh1+t1k1−h1k1,h2+t2k2−h2k2=minBPh1+t1k1−h1k1,BQh2+t2k2−h2k2≥minBPt1,BQt2=BP×Qt1,t2.
Also,(41)AP×Qh1,h2+t1,t2−k1,k2−h1,h2k1,k2=AP×Qh1+t1k1−h1k1,h2+t2k2−h2k2=maxAPh1+t1k1−h1k1,AQh2+t2k2−h2k2≤maxAPt1,AQt2=AP×Qt1,t2,FP×Qh1,h2+t1,t2−k1,k2−h1,h2k1,k2=FP×Qh1+t1k1−h1k1,h2+t2k2−h2k2=maxFPh1+t1k1−h1k1,FQh2+t2k2−h2k2≤maxFPt1,FQt2=FP×Qt1,t2.
Therefore,P×Q is a PFI of E1×E2.


## 5. Homomorphism of Picture Fuzzy Ideals of Near-Rings

This section is concerned with the direct product of PFIs of a NR.Definition 11.
LetR and S be two NRs. Then, the mapping f:R⟶S is called a near-ring homomorphism (NRH) if for all h,k∈R, the following hold:(1)
fh+k=fh+fk.(2)
fhk=fhfk.Definition 12.
LetU and Y be two nonempty sets and f:U⟶Y be a function.(1)
IfQ is a PFS in Y, then the preimage of Q under f, denoted by f−1Q, is the PFS in U defined by f−1Q=h,f−1BQh,f−1AQh,f−1FQh:h∈U,wheref−1BQh=BQfh, and so on.(2)
IfPis a PFS in U, then the image of P under f, denoted by fP, is the PFS in Y defined by fP=k,fBPk,fAPk,fFPk:k∈Y,where(42)fBPk=Suph∈f−1kBPh,iff−1k≠0E,0,otherwise,fAPk=Suph∈f−1kAPh,iff−1k≠0E,0,otherwise,fFPk=Suph∈f−1kFPh,iff−1k≠0E,0,otherwise,wherefFPk=1−f1−FPk.Theorem 5.
LetE and E′ be two NRs and f be a homomorphism of E onto E′. If P′ is a PFI of E′, then f−1P is a PFI of E.Proof.
Supposeh,k,t∈E. Then, we can deduce the following inequalities:
For membership grade,(43)f−1BPh−k=BPfh−k=BPfh−fk≥minBPfh,BPfk=minf−1BPh,f−1BPk.
Similarly, for abstinence grade, we obtain(44)f−1APh−k=APfh−k=APfh−fk≤maxAPfh,APfk=maxf−1APh,f−1APk,and for falsity grade, we write(45)f−1FPh−k=FPfh−k=FPfh−fk≤maxFPfh,FPfk=maxf−1FPh,f−1FPk.
Also, we acquire the following:(46)f−1BPk+h−k=BPfk+h−k=BPfk+fh−fk≥minBPfh=minf−1BPh.
Similarly,(47)f−1APk+h−k=APfk+h−k=APfk+fh−fk≤maxAPfh=maxf−1APh,f−1FPk+h−k=FPfk+h−k=FPfk+fh−fk≤maxFPfh=maxf−1FPh.
Furthermore, for truth grade, we obtain(48)f−1BPhk=BPfhk=BPfhfk≥minBPfk=minf−1BPk.
Similarly, for abstinence grade, we get(49)f−1APhk=APfhk=APfhfk≤maxAPfk=maxf−1APk,and for falsity, we note(50)f−1FPhk=FPfhk=FPfhfk≤maxFPfk=maxf−1FPk.
Finally, for truth grade, we obtain(51)f−1BPh+tk−hk=BPfh+tk−hk=BPfh+ftfk−fhfk≥minBPft=minf−1BPt.
Similarly, for abstinence grade,(52)f−1APh+tk−hk=APfh+tk−hk=APfh+ftfk−fhfk≤maxAPft=maxf−1APt,and for falsity grade,(53)f−1FPh+tk−hk=FPfh+tk−hk=FPfh+ftfk−fhfk≤maxFPft=maxf−1FPt.
Therefore,f−1P is a PFI of E.Theorem 6.
LetE1 and E2 be two NRs and f be a homomorphism of E1 and E2. If A is a PFI of E1, then fP is a PFI of E2.Proof.
Letk1,k2,k3∈E2 and h1,h2,h1∈E1. Then, the following are observed.
For truth grade, we can write(54)fBPk1−k2=suph1,h2∈f−1E2BPh1−h2≥suph1,h2∈f−1E2minBPh1,BPh2=minsuph1∈f−1E2BPh1,suph2∈f−1E2BPh2=minfBPk1,fBPk2.
Also,(55)fBPk1+k2−k1=suph1,h2∈f−1E2BPh1+h2−h1≥suph1∈f−1E2BPh1=fBPk1.
Furthermore, we write(56)fBPk1k2=suph1,h2∈f−1E2BPh1h2≥suph2∈f−1E2BPh2=fBPk2.
Finally, we get(57)fBPk1+k3k2−k1k2=suph1,h2,h3∈f−1E2BPh1+h3h2−h1h2≥suph1,h2,h3∈f−1E2BPh3=fBPk3.
Similarly, we can deduce for abstinence grade that(58)fAPk1−k2≤maxfAPk1,fAPk2,fAPk1+k2−k1≤fAPk2,fAPk1k2≤fAPk2,fAPk1+k3k2+k1k2≤fAPk3,and for falsity grade, we deduce that(59)fFPk1−k2≤maxfFPk1,fFPk2,fFPk1+k2−k1≤fFPk2,fFPk1k2≤fFPk2,fFPk1+k3k2+k1k2≤fFPk3.
Hence,fP is a PFI of E2.
Note that, by choosing the value of nonmembership as zero, the main results of the article are obtained in the setting of fuzzy sets. However, taking the value of abstinence as zero, we acquire the mentioned results in intuitionistic fuzzy sets. These are the extensions of the presented work.


## 6. Conclusion

The theory of a picture fuzzy set is basically the extension of the fuzzy set, containing the grade of membership, the grade of abstinence, and the grade of nonmembership with a rule, provided the sum of all grades cannot exceed the unit interval. Keeping in mind the advantages of the picture fuzzy set, in this manuscript, we presented the PFSNR and PFIs of NRs and explained some of their algebraic properties. We have proved that the union of two PFIs of a NR is a PFI of that NR. Also, we have proved that the positive integral powers of a PFI of a NR are a PFI. We have defined the direct product on PFIs of NRs and proved that the direct product of any two PFIs of NRs is a PFI. Moreover, we showed that we can extend the result for a finite number of PFIs of NRs.In future, we will investigate the decision-making problems based on Pythagorean fuzzy sets [17, 18]. An investigation will be carried out about the Pythagorean fuzzy near-rings and Pythagorean ideals of a near-ring, and their algebraic properties will be discussed.



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