# A New Approach to Evaluate Regular Semirings in terms of Bipolar Fuzzy k-Ideals Using k-Products

Authors: Shahida Bashir; Ahmad N. Al-Kenani; Tanzila Kausar; Rabia Mazhar
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we provide a generalized form of ideals that isk-ideals of semirings with the combination of a bipolar fuzzy set (BFS). The BFS is a generalization of fuzzy set (FS) that deals with uncertain problems in both positive and negative aspects. The main theme of this paper is to present the idea of α,β-bipolar fuzzy k-subsemiring (k-BFSS), α,β-bipolar fuzzy k-ideals (k-BFIs), and α,β-bipolar fuzzy k-bi-ideals (k-BFbIs) in semirings by applying belongingness ∈ and quasi-coincidence q of the bipolar fuzzy (BF) point. After that, upper and lower parts of k-product of BF subsets of semirings are introduced. Lastly, the notions of k-regular and k-intraregular semirings in terms of ∈,∈∨q-k–BFIs and ∈,∈∨q-k–BFbIs are characterized.


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## Body

## 1. Introduction and Motivation

In engineering, decision making theory, management science, and medical science, we may face uncertainty and vagueness in the data. In classical mathematics, all the formulae and methods consist on crisp set that cannot tackle the vague problems. After an extensive struggle, many theories are invented to tackle such problems. In 1965, Zadeh [1] gave the idea of fuzzy set theory to handle such complicated problems. This concept is applied on theory of rings, theory of groups, real analysis, logics, and topological space. In FS, membership degree is limited to 0,1, but there was a difficulty to deliberate the irrelevancy of data. To resolve this difficulty, Zhang [2] offered the idea of BFS in which the membership degree is −1,1. Since the BFS theory is a development of the FS theory, thus a BFS in semirings is also useful.Bipolarity is a significant theory which is mostly used in real life. It is noticed that people may have different responses at a time for the same qualities of an item or a plan. Someone may have negative thoughts and the other one may have positive thoughts. For example,$100 is a big amount for a needy person but at the same phase, this amount is negligible for a rich man. BFS is also useful in database query, psychology, image processing, multicriteria decision making, argumentation, and in human reasoning problems. Recently, Riaz et al. [3] have worked on BFS with the combination of picture fuzzy set and discussed its application on pattern recognition. Facial recognition or face detection is an artificial intelligence-based computer technology used to find and identify human faces in digital images; see [4–15] for more examples and results which are relevant to BFS.The role of semirings as an algebraic tool is very important in theoretical computer science ([16, 17]). Semirings are broadly used in formal languages, automata theory, optimization theory, graph theory, coding theory, and theory of discrete event dynamical systems. In the structural theory, ideals play a central role and are beneficial for many other purposes. In general, ideals of semiring do not coincide with ideals of the ring. Many results in rings apparently are not equivalent to results in semirings using only ideals. Henriksen [18] introduced a most generalized form of ideals in semiring, which is k-ideal, with the characterization that if the semiring R is a ring, then a complex in R is a k-ideal equivalent to a ring ideal. The k-ideal is the most restricted form of an ideal. Every k-ideal is ideal but the converse does not hold. The set of whole numbers  is a semiring. Let I=0,2∪3,4,5,… then I is an ideal of  while it is not a k-ideal of  because 2,2+1∈Ibut1∉I If we take a subset I=0,3,6,9,… of , then I is a k-ideal of . For more details; see [19].


### 1.1. Related Works

Zadeh’s fuzzy set is a much innovative, crucial, and useful set due to its significance in multiple research dimensions. The fuzzy set addresses the ill-defined situations by which we are often encountered. From these ill-defined situations, we can evaluate results by using degree of membership of fuzzy set but the bipolar fuzzy set is much better set to manage uncertainty, vagueness, and impreciseness than a fuzzy set. The bipolar fuzzy set is much valuable due to its degree of membership [-1,1].


### 1.2. Applications of the Proposed Model

Bipolar fuzzy ideals are very suitable in the field of image processing, database query, human reasoning, psychology, argumentation, multi-criteria decision making, etc. There are extensive applications of semirings in several fields, such as formal languages, graph theory, automata theory, generalized fuzzy computation, discrete theory, and many branches of applied mathematics. The combination of bipolar fuzzy ideals with semirings is a very useful tool for providing languages to deal with complicated problems in information science.


### 1.3. Innovative Contribution

In fuzzy sets, the degree of membership was restricted to [0,1]. In our realistic life, someone may have a negative response and other one may have a positive response at a time for the same characteristic of an object. In this regard, bipolarity is a very useful concept, which is commonly used in our real world problems. In the automata theory, many applications of fuzzy semirings are given by Ahsan et al. [20]. Lee [21, 22] introduced BFIs of BCK\BCI-algebras. Jun and Kavikumar [8] describe the idea of a BF finite state machine. Majuder [23] studied characteristics of bipolar fuzzy 1,2-ideals in semigroups. Zhou and Li ([24, 25]) utilized BFS in semirings. Manemaran and Chellappa [26] utilized BFSs in the group theory. Bhakat and Das ([27, 28]) initiated the idea of α,β-fuzzy subgroups and extension of this work is α,β-fuzzy ideals of hemirings introduced by Dudek et al. [29]. Shabir et al. [30] gave the idea of characterizations of semigroups by α,β-fuzzy ideals. In [31], Dawaz studied ∈,∈∨q-fuzzy subnear-rings. Ibrar et al. [32] characterized regular ordered semigroups by using α,β bipolar fuzzy generalized bi ideals. In [33], Shabir et al. worked on α,β–BFSSs and α,β–BFIs of semirings. Recently, Shabir et al. [34] studied BF hyperideals in regular semihypergroups. Here, we have extended the above work for α,β-k–BFSSs and α,β–k–BFIs of semirings. See [35–39] for more details.


### 1.4. Organization of the Paper

In Section1, we have familiarized with the physical background of the bipolar fuzzy semiring and its characterization toward regular and intraregular semirings. Section 2 describes the literature review of k-BFSS, k-BFI, and some new important definitions which will be used as the ground work for the major works. In Section 3, the concepts of α,β–k–BFSS and α,β–k-BFI of semirings are discussed. In Section 4, we have worked on upper and lower parts of BFSs. In Section 5, the description of theorems of k-regular and k-intraregular semirings in terms of ∈,∈∨q-k–BFIs is given. In Section 6, the comparative study and the last section consist of the conclusions and future plans.The list of acronyms used here is given in Table1.Table 1
List of acronyms.
AcronymsRepresentationFSFuzzy setBFBipolar fuzzyBFSBipolar fuzzy setk-BFSSBipolar fuzzyk-subsemiringk-BFIBipolar fuzzyk-idealk-BFbIBipolar fuzzyk-bi-idealIffif and only if


## 1.1. Related Works

Zadeh’s fuzzy set is a much innovative, crucial, and useful set due to its significance in multiple research dimensions. The fuzzy set addresses the ill-defined situations by which we are often encountered. From these ill-defined situations, we can evaluate results by using degree of membership of fuzzy set but the bipolar fuzzy set is much better set to manage uncertainty, vagueness, and impreciseness than a fuzzy set. The bipolar fuzzy set is much valuable due to its degree of membership [-1,1].


## 1.2. Applications of the Proposed Model

Bipolar fuzzy ideals are very suitable in the field of image processing, database query, human reasoning, psychology, argumentation, multi-criteria decision making, etc. There are extensive applications of semirings in several fields, such as formal languages, graph theory, automata theory, generalized fuzzy computation, discrete theory, and many branches of applied mathematics. The combination of bipolar fuzzy ideals with semirings is a very useful tool for providing languages to deal with complicated problems in information science.


## 1.3. Innovative Contribution

In fuzzy sets, the degree of membership was restricted to [0,1]. In our realistic life, someone may have a negative response and other one may have a positive response at a time for the same characteristic of an object. In this regard, bipolarity is a very useful concept, which is commonly used in our real world problems. In the automata theory, many applications of fuzzy semirings are given by Ahsan et al. [20]. Lee [21, 22] introduced BFIs of BCK\BCI-algebras. Jun and Kavikumar [8] describe the idea of a BF finite state machine. Majuder [23] studied characteristics of bipolar fuzzy 1,2-ideals in semigroups. Zhou and Li ([24, 25]) utilized BFS in semirings. Manemaran and Chellappa [26] utilized BFSs in the group theory. Bhakat and Das ([27, 28]) initiated the idea of α,β-fuzzy subgroups and extension of this work is α,β-fuzzy ideals of hemirings introduced by Dudek et al. [29]. Shabir et al. [30] gave the idea of characterizations of semigroups by α,β-fuzzy ideals. In [31], Dawaz studied ∈,∈∨q-fuzzy subnear-rings. Ibrar et al. [32] characterized regular ordered semigroups by using α,β bipolar fuzzy generalized bi ideals. In [33], Shabir et al. worked on α,β–BFSSs and α,β–BFIs of semirings. Recently, Shabir et al. [34] studied BF hyperideals in regular semihypergroups. Here, we have extended the above work for α,β-k–BFSSs and α,β–k–BFIs of semirings. See [35–39] for more details.


## 1.4. Organization of the Paper

In Section1, we have familiarized with the physical background of the bipolar fuzzy semiring and its characterization toward regular and intraregular semirings. Section 2 describes the literature review of k-BFSS, k-BFI, and some new important definitions which will be used as the ground work for the major works. In Section 3, the concepts of α,β–k–BFSS and α,β–k-BFI of semirings are discussed. In Section 4, we have worked on upper and lower parts of BFSs. In Section 5, the description of theorems of k-regular and k-intraregular semirings in terms of ∈,∈∨q-k–BFIs is given. In Section 6, the comparative study and the last section consist of the conclusions and future plans.The list of acronyms used here is given in Table1.Table 1
List of acronyms.
AcronymsRepresentationFSFuzzy setBFBipolar fuzzyBFSBipolar fuzzy setk-BFSSBipolar fuzzyk-subsemiringk-BFIBipolar fuzzyk-idealk-BFbIBipolar fuzzyk-bi-idealIffif and only if


## 2. Preliminaries

Basic definitions, results, and some back ground material are given. Here, a subset means a nonempty subset.Thek-closure of a subset K of a semiring R is K¯=u∈R|∃c,d∈Ksuch thatu+c=d. A subsemiring K of R is called k-subsemiring of R if and only if (iff) K=K¯. Thus, an ideal (left, right) K of R is a k-ideal (left, right) if K=K¯. A subsemiring K is called a k-bi-ideal of R if KRK⊆K and K=K¯. A semiring R is known as k-regular if for all, u∈R there are elements c,c′∈R such as u+ucu=uc′u. A semiring R is known as k-intraregular if for all u∈R, there are elements ai,aj′,bi,bj′∈R such that u+∑i=1maiu2bi=∑j=1naj′u2bj′ [40].Lemma 1 (see [40]).
A semiringR is k-regular iff K∩P, =KP¯ for every right k-ideal K and left k-ideal P of R.Lemma 2 (see [40]).
A semiringR is k-regular iff P=PRP¯ for every k-bi-ideal P of R.Lemma 3 (see [40]).
A semiringR is k-intraregular iff K∩P⊆KP¯ for every left k-ideal K and every right k-ideal P of R.
Now, we will discuss some essential concepts related to BFS.
A BF subsetΨ=R;Ψn,Ψp of a semiring R, where Ψn:R⟶−1,0 gives the satisfaction degree for somewhat opposite property of Ψ and Ψp:R⟶0,1 gives the degree of satisfaction of the corresponding property of Ψ. While in a fuzzy set, the membership function maps to 0,1. The difference between a fuzzy set and bipolar fuzzy set is shown by the following example.
LetA=u,v,w,x,y,z be a set of workers of a company. Define a fuzzy set on A with fuzzy property “honesty.” The workers v,xandy having property “honesty” mapped to [0,1] as shown by a bar graph in Figure 1. While other workers have no membership degree in range [0,1] as they are not honest. In the fuzzy set, we can cover only a positive aspect of any situation. We cannot deal with negative aspects of the situation. To facilitate, we deal such problems with a bipolar fuzzy set. The property “dishonesty” is against “honesty”. The workers u,wandz are mapped to [-1,0] with property “dishonesty.” In such a way, the bipolar fuzzy set gives information about all elements as shown in Figure 2.
Foru∈R, if Ψpu=0, Ψnu≠0 then u does not hold the property of Ψ and if Ψpu≠0, Ψnu=0 then u holds the property of Ψ If Ψpu≠0 and Ψnu≠0, then its opposite property and membership function intersect. A BF subset 0=R:0n,0p is defined by 0nu=0=0pu for any u∈R
LetΨ=R;Ψn,Ψp and μ=R;μn,μp be two BF subsets of a semiring R. Then, Ψ≤μ if Ψnu≥μnu and Ψpu≤μpu for all u∈R and Ψ=μ if Ψ≤μ and μ≤Ψ. The intersection and union of Ψ and μ is defined by Ψ∩μ=R;Ψn∨μn,Ψp∧μp and Ψ∪μ=R;Ψn∧μn,Ψp∨μp, respectively. Throughout the paper, R is semiring unless otherwise identified.Figure 1
Fuzzy set.Figure 2
Bipolar fuzzy set.Definition 1.
LetΨ=R;Ψn,Ψp and μ=R;μn,μp be two BF subsets of R. The k-product Ψ∘kμ=R;Ψn∘kμn,Ψp∘kμp of R is defined as follows:(1)Ψn∘kμnu=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′,0ifuis not representable asu+∑i=1naibi=∑j=1maj′bj′,Ψp∘kμpu=∨u+∑i=1naibi=∑j=1maj′bj′∧i=1nΨpai∧∧i=1nμpbi∧∧j=1mΨpaj′∧∧j=1mμpbj′,0ifuis not representable asu+∑i=1naibi=∑j=1maj′bj′,for u,ai,bi∈R.Proposition 1.
AssumeΨ,μ,ν,σ are BF subsets of R. At that time, Ψ≤ν and μ≤σ; this implies that Ψ∘kμ≤ν∘kσ.Proof.
Letu∈R then Ψp∘kμpu=∨u+∑i=1naibi=∑j=1maj′bj′∧i=1nΨnai∧∧i=1nμnbi∧∧j=1mΨnaj′∧∧j=1mμnbj′
AsΨp≤νpimpliesΨpai≤νpai, Ψpaj′≤νpaj′
Andμp≤σpmeansμpbi≤σpbi, μpbj′≤σpbj′
So,Ψp∘kμpu≤∨u+∑i=1naibi=∑j=1maj′bj′∧i=1nνnai∧∧i=1nσnbi∧∧j=1mνnaj′∧∧j=1mσnbj′=νp∘kσpu
And,Ψn∘kμnu=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′
AsΨn≥νnimpliesΨnai≥νnai, Ψnaj′≥νnaj′
Moreover,μn≥σnmeansμnbi≥σnbi, μnbj′≥σnbj′
So,Ψn∘kμnu≥∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nνnai∨∨i=1nσnbi∨∨j=1mνnaj′∨∨j=1mσnbj′=νn∘kσnu
Hence,Ψ∘kμ≤ν∘kσ.Definition 2.
IfK is a k-subset of R Then, the bipolar characteristic function of K is denoted by CK=R;CnK,CpK and is defined by CnKu=−1ifu∈K0ifu∉K, CpKu=1ifu∈K0ifu∉K
For allu∈R, if K=R, then we have a BF subset R=R;Rn,Rp defined as R⌣nx=−1 and R⌣px=1 for all u∈RDefinition 3.
A BF subsetΨ=R;Ψn,Ψp of R is called k-BFSS of R if it satisfies the following:(i)
Ψn0≤Ψnc and Ψp0≥Ψpc;(ii)
Ψnc+d≤maxΨnc,Ψnd and Ψpc+d≥minΨpc,Ψpd;(iii)
Ψncd≤maxΨnc,Ψnd and Ψpcd≥minΨpc,Ψpd;(iv)
Ifu+c=d then Ψnu≤maxΨnc,Ψnd and Ψpu≥minΨpc,Ψpd for all u,c,d∈R.Definition 4.
A BF subsetΨ=R;Ψn,Ψp of R is said to be k-BFI (right resp. left) of R if it satisfies the following:(i)
Ψnc+d≤maxΨnc,Ψnd and Ψpc+d≥minΨpc,Ψpd;(ii)
Ψncd≤Ψnc and Ψpcd≥Ψpc, resp⋅Ψncd≤ΨndandΨpcd≥Ψpd;(iii)
Ifu+c=d then Ψnu≤maxΨnc,Ψnd and Ψpu≥minΨpc,Ψpd for all u,c,d∈R.Example 1.
A BF subsetΨ of a semiring Ⱳ =0,1,2,3,… is defined as follows:(2)Ψnu=0,ifu=1,−0.5,otherwise,Ψpu=0,ifu=1,0.5,otherwise,then Ψ is a BFI of Ⱳ but not a k-BFI because if we take 1+2=3 then Ψn1=0≥−0.5=maxΨn2,Ψn3 and Ψp1=0≤0.5=minΨp2,Ψp3.Example 2.
Let a BF subsetΨ of a semiring Ⱳ =0,1,2,3,… defined as follows:(3)Ψnu=−0.1,ifuis odd,−0.5,ifuis even,−1,ifu=0,Ψpu=0.1,ifuis odd,0.5,ifuis even,1,ifu=0,Then, it is easy to check that Ψ is a k-BFI of Ⱳ.Definition 5.
A BF subsetΨ=R;Ψn,Ψp of R is said to be bipolar fuzzy k-BFbI of R if it satisfies the following:(i)
Ψn0≤Ψnc and Ψp0≥Ψpc;(ii)
Ψnc+d≤maxΨnc,Ψnd and Ψpc+d≥minΨpc,Ψpd;(iii)
Ψncd≤maxΨnc,Ψnd and Ψpcd≥minΨpc,Ψpd;(iv)
Ψncde≤maxΨnc,Ψne and Ψpcde≥minΨpcΨpe;(v)
Ifu+c=d then Ψnu≤maxΨnc,Ψnd and Ψpu≥minΨpc,Ψpd for all u,c,d,e∈R.Lemma 4.
IfK is k-ideal (left, right) of R, then CK=R,CnK,CpK is k-BFI (left, right) of R.Proof.
We prove it for rightk–ideal. For this, we have to verify the below three inequalities, for each element c,d,e,u,v∈R.(i)
CnKc+d≤maxCnKc,CnKd and CpKc+d≥minCpKc,CpKd;(ii)
CnKcd≤, CnKc and CpKcd≤CpKc;(iii)
u+c=d⟹CnKu≤maxCnKc,CnKd and CpKu≥minCpKc,CpKd.
For the proofs of (i) and (ii), see [32]. Here, we just prove part (iii). For this, we discuss the following cases.Case 1.
Supposeu,c,d∈K
Then,CnKu=−1=CnKc=CnKd and CpKu=1=CpKc=CpKd
SinceK is a k-ideal of R, for u,c,d∈K, CnKu=−1≤maxCnKc,CnKd and CpKu=1≥minCpKc,CpKdCase 2.
Ifu,c,d∉K
Then,CnKu=0=CnKc=CnKd and CpKu=0=CpKc=CpKd, CnKu=0≤maxCnKc,CnKd and CpKu=0≥minCpKc,CpKdCase 3.
If one of thec or d does not belong to K, say c∉K
Then,CnKc=0 and CpKc=0, CnKu≤0=maxCnKc,CnKd and CpKu≥0=minCpKc,CpKd
So,CK is a k-BFI.Lemma 5.
IfK is a k-bi-ideal of R, thenCK=R,CnK,CpK is k-BFbI of R.Proof.
Straightforward.


## 3.α,β Bipolar Fuzzy k-Ideals

Here, ideas ofα,β-k–BFSS, α,β-k–BFI, and α,β-k–BFbI of semirings are introduced and their related properties are discussed.Definition 6 (see [32,33]).
Letc,d∈−1,0×0,1, then u/c,d is called a BF point in R if the BF subset Ψ of R is given in the form of the following:(4)Ψpa=d,ifa=u,0,ifa≠u,Ψna=c,ifa=u,0,ifa≠u.for all a,u∈R.Definition 7 (see [32, 33]).
For a BF subsetΨ of R and a BF point u/c,d, we say that u/c,d belongs to (quasi-coincident to resp.) the Ψ is written as u/c,d∈Ψ, u/c,dqΨresp. if it satisfies the following:(i)
u/c,d∈Ψ if Ψnu≤c and Ψpu≥d(ii)
u/c,dqΨ if Ψnu+c<−1 and Ψpu+d>1(iii)
u/c,d∈∨qΨ if u/c,d∈Ψ or u/c,dqΨ(iv)
u/c,d∈∧qΨ if u/c,d∈Ψ and u/c,dqΨ
Here, we considerα,β∈∈,q,∈∨q,∈∧q and α≠∈∧q. Consider a BF subset Ψ=R;Ψn,Ψp such that Ψnu≥−0.5 and Ψpu≥0.5 for all u∈R and u/c,d∈∧qΨ. Then, Ψnu≤c, Ψpu≥d, Ψnu+c<−1 and Ψpu+d>1. It follows that −1>Ψnu+c≥Ψnu+Ψnu=2Ψnu, which implies that Ψnu≤−0.5, thus a contradiction arises. So, u/c,d∈∧q¯Ψ, and hence, α≠∈∧q.Definition 8.
A BF subsetΨ=R;Ψn,Ψp of R is called an α,β-k–BFSS of R if for all t,u∈R and c1,d1,c2,d2∈−1,0×0,1, it satisfies the following:(i)
t/c1,d1αΨ and u/c2,d2αΨ⟶t+u/c1∨c2,d1∧d2βΨ;(ii)
t/c1,d1αΨ and u/c2,d2αΨ⟶tu/c1∨c2,d1∧d2βΨ;(iii)
Supposeu,c,d∈R such as u+c=d, then c/c1,d1αΨ and d/c2,d2αΨ⟶u/c1∨c2,d1∧d2βΨ here, α≠,∈∧q.Definition 9.
A BF subsetΨ=R;Ψn,Ψp of R is called an α,β-k–BFI (left, resp. right) of R, if for c1,d1,c2,d2∈−1,0×0,1, and for all t,u∈R, if it satisfies the following:(i)
t/c1,d1αΨ and u/c2,d2αΨ⟶t+u/c1∨c2,d1∧d2βΨ;(ii)
t/c1,d1αΨ and u∈R⟶ut/c1,d1βΨ (resp. tu/c1,d1βΨ;(iii)
Supposeu,c,d∈R such as u+c=d, then c/c1,d1αΨ and d/c2,d2αΨ⟶u/c1∨c2,d1∧d2βΨ; here, α≠∈∧q.Definition 10.
A BF subsetΨ=R;Ψn,Ψp of R is called an α,β-k-BFbI of R, if for c1,d1,c2,d2∈−1,0×0,1, and for every t,u,c,d∈R it satisfies the following:(i)
t/c1,d1αΨ and u/c2,d2αΨ⟶t+u/c1∨c2,d1∧d2βΨ;(ii)
t/c1,d1αΨ and u/c2,d2αΨ⟶tu/c1∨c2,d1∧d2βΨ;(iii)
t/c1,d1αΨ and v/c3,d3αΨ⟶tuv/c1∨d3,c1∨d3βΨ;(iv)
Supposeu,c,d∈R such as u+c=d, then c/c1,d1αΨ and d/c2,d2αΨ⟶u/c1∨c2,d1∧d2βΨ; here, α≠∈∧q.Theorem 1.
If a BF subsetΨ=R;Ψn,Ψp of R is ∈,∈∨q-k-BFSS of R, then it fulfills the following conditions for each c,d,u∈R.(i)
Ψnc+d≤maxΨnc,Ψnd,−0.5 and Ψpc+d≥minΨpc,Ψpd,0.5(ii)
Ψncd≤maxΨnc,Ψnd,−0.5 and Ψpcd≥minΨpc,Ψpd,0.5(iii)
Foru+c=d,Ψnu≤maxΨnc,Ψnd,−0.5 and Ψpu≥minΨpc,Ψpd,0.5Proof.
For the proofs of part (i) and part (ii), see [33]; here, we just prove part (iii).
Forc,d∈R, four following cases arise:(i)
maxΨnc,Ψnd≤−0.5 and minΨpc,Ψpd≥0.5(ii)
maxΨnc,Ψnd≤−0.5 and minΨpc,Ψpd<0.5(iii)
maxΨnc,Ψnd>−0.5 and minΨpc,Ψpd≥0.5(iv)
maxΨnc,Ψnd>−0.5 and minΨpc,Ψpd<0.5
We will prove with contrary supposition. Then, assume forc,d∈R, Ψnu>maxΨnc,Ψnd,−0.5 or Ψpu<minΨpc,Ψpd,0.5.Case 1.
maxΨnc,Ψnd≤−0.5 and minΨpc,Ψpd≥0.5, then Ψnu>−0.5 or Ψpu<0.5 implies u/−0.5,0.5∈¯Ψ. Now, again consider maxΨnc,Ψnd≤−0.5, then Ψnc≤−0.5,Ψnd≤−0.5 and minΨpc,Ψpd≥0.5 then Ψpc≥0.5,Ψpd≥0.5. Therefore, c/−0.5,0.5∈Ψ,, d/−0.5,0.5∈Ψ but for u∈R,u/−0.5,0.5∈¯, Ψ. Similarly, u/−0.5,0.5q¯, Ψ because Ψnu−0.5>−1 or Ψpu+0.5<1. So, u/−0.5,0.5∈∨q¯, Ψ, which is a contradiction.Case 2.
maxΨnc,Ψnd≤−0.5 and minΨpc,Ψpd<0.5, then Ψnu>−0.5 or Ψpu<minΨpc,Ψpd, so for t∈0,0.5, Ψpu<t=minΨpc,Ψpd implies u/−0.5,t∈¯, Ψ. Now, again consider maxΨnc,Ψnd≤−0.5, then Ψnc≤−0.5 and Ψnd≤−0.5. Also, minΨpc,Ψpd=t implies Ψpc≥t, and Ψpd≥t. Then, we get c/−0.5,t∈Ψ and d/−0.5,t∈Ψ but for u∈R,u/−0.5,t∈¯, Ψ. Similarly, u/−0.5,tq¯, Ψ because Ψpu+t<t+t<1. Thus, u/−0.5,t∈∨q¯, Ψ, which contradicts our supposition.Case 3.
maxΨnc,Ψnd>−0.5 and minΨpc,Ψpd≥0.5, then Ψnu>maxΨnc,Ψnd or Ψpu<0.5. So, for s∈−0.5,0, Ψnu>s=maxΨnc,Ψnd implies u/s,0.5∈¯, Ψ. Moreover, maxΨnc,Ψnd=s implies Ψnc≤s and Ψnd≤s. Also, minΨpc,Ψpd≥0.5 indicates Ψpc≥0.5 and Ψpd≥0.5. Then, we get c/s,0.5∈, Ψ,d/s,0.5∈, Ψ but for u∈R,u/s,0.5∈¯, Ψ. Similarly, u/s,0.5q¯, Ψ because Ψnu+s>s+s>−1. Therefore, u/s,0.5∈∨q¯, Ψ which is a contradiction.Case 4.
maxΨnc,Ψnd>−0.5 and minΨpc,Ψpd<0.5, then Ψnu>maxΨnc,Ψnd or Ψpu<minΨpc,Ψpd. Then for s∈−0.5,0 and t∈0,0.5, Ψnu>s=maxΨnc,Ψnd and Ψpu<t=minΨpc,Ψpd then u/s,t∈¯, Ψ.
Also,maxΨnc,Ψnd=s implies Ψnc≤s and Ψnd≤s. Moreover, minΨpc,Ψpd=t implies Ψpc≥t and Ψpd≥t. Now, combining all these, we get c/s,t∈Ψ, ,d/s,t∈Ψ.
However, foru∈R,u/s,t∈¯, Ψ. Similarly, u/s,tq¯, Ψ because Ψnu+s>s+s>−1 or Ψpu+t<t+t<1. Thus, u/s,t∈∨q¯, Ψ which is a contradiction.
Therefore,Ψnu≤maxΨnc,Ψnd,−0.5 and Ψpu≥minΨpc,Ψpd,0.5 for each c,d,u∈R. The reverse of this theorem does not hold in general as shown by the following example.Example 3.
LetR=α,β,δ be a semiring under operations defined as below:(see Tables 2 and 3)
LetΨ=R;Ψn,Ψp be a BF subset defined by Ψnα=−0.48,Ψnβ=−0.3,Ψnδ=−0.2, and Ψpα=0.55,Ψpβ=0.9,Ψpδ=0.72.
Then, foru+c=d,, Ψnu≤maxΨnc,Ψnd,−0.5 and Ψpu≥minΨpc,Ψpd,0.5 hold for all c,d,u∈R. Now, take α+δ=δ, then δ/−0.2,0.72∈Ψ and δ/−0.2,0.72∈Ψ but α/−0.2,0.72∈¯Ψ as Ψpα=0.55<0.72. Also, Ψnα=−0.48+−0.2=−0.68>−1 implies α/−0.2,0.72q¯Ψ. So, α/−0.2,0.72∈∨q¯Ψ.Table 2
Binary addition.
+αβδααβδββββδδβδTable 3
Binary multiplication.
∗αβδααααβαααδαααTheorem 2.
If a BF subsetΨ=R;Ψn,Ψp of R is (∈, ∈∨q)-k-BFbI of R, then it fulfills the following conditions for each c,d,u∈R(i)
Ψnc+d≤maxΨnc,Ψnd,−0.5 and Ψpc+d≥minΨpc,Ψpd,0.5;(ii)
Ψncd≤maxΨnc,Ψnd,−0.5 and Ψpcd≥minΨpc,Ψpd,0.5;(iii)
Ψncde≤maxΨnc,Ψne,−0.5 and Ψpcde≥minΨpc,Ψpe,0.5;(iv)
Foru+c=d,, Ψnu≤maxΨnc,Ψnd,−0.5 and Ψpu≥minΨpc,Ψpd,0.5.Proof.
Proof follows from Theorem1.Theorem 3.
If a BF subsetΨ=R;Ψn,Ψp of R is an ∈,∈∨q-kBFI (left, resp. right) of R, then it fulfills the following statements for each c,d,u∈R(i)
Ψnc+d≤maxΨnc,Ψnd,−0.5 and Ψpc+d≥minΨpc,Ψpd,0.5;(ii)
Ψncd≤maxΨnd,−0.5 and Ψpcd≥minΨpd,0.5 (resp. Ψncd≤maxΨnc,−0.5 and Ψpcd≥minΨpc,0.5);(iii)
Foru+c=d,, Ψnu≤maxΨnc,Ψnd,−0.5 and Ψpu≥minΨpc,Ψpd,0.5.Proof.
Proof follows from Theorem1.


## 4. Upper and Lower Components of a Bipolar Fuzzyk-Set

In this part, definitions and some lemmas relating to the upper and lower parts of ak-BFS of R are studied. These terms will be useful in the next section.Definition 11.
LetΨ=R,Ψn,Ψp be a k-BFS of R. Then, for u∈R, upper part Ψ+=R,Ψn+,Ψp+ of Ψ is written as Ψn+u=Ψnu∧−0.5, Ψp+u=Ψpu∨0.5 and lower part Ψ=R,Ψn,Ψp of Ψ is written as Ψn−u=Ψnu∨−0.5 , Ψp−u=Ψpu∧0.5.Lemma 6.
LetΨ=R,Ψn,Ψp and R,μn,μp be two k-BFSs in R, then (i) Ψ∧μ−=Ψ−∧μ−; (ii) Ψ∨μ−=Ψ−∨μ−; (iii) Ψ∘kμ−=Ψ−∘kμ−.Proof.
Straightforward.Lemma 7.
LetΨ=R,Ψn,Ψp and R,μn,μpbe two k-BFSs in R, then (i) Ψ∧μ+=Ψ+∧μ+; (ii) Ψ∨μ+=Ψ+∨μ+; (iii) Ψ∘kμ+=Ψ+∘kμ+.Proof.
Straightforward.Lemma 8.
For an∈,∈∨q-k–BFIRΨ=R,Ψn,Ψp also ∈,∈∨q-k–BFIL μ=R,μn,μp, we have Ψokμ−≤Ψ∧μ−Proof.
Foru∈R, if Ψ∘kμu=0 that is Ψn∘kμnu=0 and Ψp∘kμpu=0 then Ψn∘kμn−u=0∨−0.5=0≥Ψn∨μn−u and Ψp∘kμp−u=0∧0.5=0≤Ψp∨μp−u.
Otherwise,(5)Ψn∘kμn−u=Ψn∘kμnu∨−0.5=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′∨−0.5=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnai∨−0.5∨∨i=1nμnbi∨−0.5∨∨j=1mΨnaj′∨−0.5∨∨j=1mμnbj′∨−0.5∨−0.5≥∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnaibi∨∨i=1nμnaibi∨∨j=1mΨnaj′bj′∨∨j=1mμnaj′bj′∨−0.5≥∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnaibi∨−0.5∨∨i=1nμnaibi∨−0.5∨∨j=1mΨnaj′bj′∨−0.5∨∨j=1mμnaj′bj′∨−0.5∨−0.5≥∧u+∑i=1naibi=∑j=1maj′bj′Ψn∑i=1naibi∨μn∑i=1naibi∨Ψn∑j=1maj′bj′∨μn∑j=1maj′bj′∨−0.5=Ψn∨μnu∨−0.5=Ψn∨μn−u.
So,Ψn∘kμn−u≥Ψn∨μn−u. Similarly, we can prove Ψp∘kμp−u≤Ψp∧μp−u.
Hence,Ψ∘kμ−≤Ψ∧μ−.Definition 12.
IfL is a k-subset of R. The upper part CL+=R;CnL+,CpL+ of the bipolar characteristic function CL=R;CnL,CpL of L is defined as CnL+x=−1ifx∈L−0.5ifx∉L, CpL+x=1ifx∈L0.5ifx∉L.
Similarly, the lower partCL−=R;CnL−,CpL− of the bipolar characteristic function CL=R;CnL,CpL of L is defined as CnL−x=−0.5ifx∈L0ifx∉L,CpL−x=0.5ifx∈L0 ifx∉L.Lemma 9.
LetPandQ be two nonempty k-subsets of R, then(i)
CP∧CQ−=CP∩Q−;(ii)
CP∨CQ−=CP∪Q−;(iii)
CP∘kCQ−=CPQ¯−.Proof.
The proofs of part (i) and (ii) are quite clear. (iii) Letx∈PQ¯. Then, CnPQ¯−x=−0.5 and CpPQ¯−x=0.5.
Givenx∈PQ¯, so there are some elements ai,aj′∈P and bi,bj′∈Q such that x+∑i=1naibi=∑j=1maj′bj′; thus, we have CnPai=−1=CnPaj′=CnQbi=CnQbj′ and CpPai=−1=CpPaj′=CpQbi=CpQbj′
Now,(6)CnP∘kCnQ−x=CnP∘kCnQx∨−0.5=∧x+∑i=1naibi=∑j=1maj′bj′∨i=1nCnPai∨∨i=1nCnQbi∨∨j=1mCnPaj′∨∨j=1mCnQbj′∨−0.5=∧x+∑i=1naibi=∑j=1maj′bj′−1∨−1∨−1∨−1∨−0.5=−0.5=CnPQ¯−x.
Also,(7)CpP∘kCpQ−x=CpP∘kCpQx∧0.5=∨x+∑i=1naibi=∑j=1maj′bj′∧i=1nCpPai∧∧i=1nCpQbi∧∧j=1mCpPaj′∧∧j=1mCpQbj′∧0.5=∧x+∑i=1naibi=∑j=1maj′bj′1∧1∧1∧1∧0.5=0.5=CpPQ¯−x.
So,CP∘kCQ−=CPQ¯−. If x∉PQ¯, then CnPQ¯−x=0 and CpPQ¯−x=0.
Then,CnP∘kCnQ−x=CnP∘kCnQx∨−0.5=0∨−0.5=0=CnPQ¯−x Moreover, CpP∘kCpQ−x=CpP∘kCpQx∧0.5=0∧0.5=CpPQ¯−x. Hence, CP∘kCQ−=CPQ¯−.Lemma 10.
IfP is k-ideal (left, resp. right) of R, then CP−=R,CnP−,CpP− is ∈,∈∨q-k–BFI (left, resp. right) of R.Proof.
Straightforward.Lemma 11.
IfP is k-bi-ideal of R, then CP−=R,CnP−,CpP− is ∈,∈∨q-k-BFbI of R.Proof.
Straightforward.


## 5. k-Regular and k-Intraregular Semirings

In this portion, many interesting theorems on∈,∈∨q-k-BFIs and ∈,∈∨q-k-BFbIs of k-regular and k-intraregular semiring are presented.Theorem 4.
For a semiringR, the given statements are equivalent.(i)
R is k-regular.(ii)
Ψokμ−=Ψ∧μ− for every ∈,∈∨q-k-BFIRΨ=R,Ψn,Ψp and ∈,∈∨q-k-BFILμ=R,μn,μp of R.Proof.
AssumeR is k-regular semiring. Let u∈R, then there are elements c,c′∈ such as u+ucu=uc′u. Moreover, we have the following equation: (8)Ψn∘kμn−u=Ψn∘kμnu∨−0.5=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′∨−0.5≤Ψnuc∨μnu∨Ψnuc′∨μnu∨−0.5≤Ψnu∨−0.5∨μnu∨Ψnu∨−0.5∨μnu∨−0.5=Ψnu∨μnu∨−0.5=Ψn∨μn−u.
Similarly,Ψp∘kμp−u≥Ψp∧μp−u. Consequently, Ψ∧μ−≤Ψ∘kμ−. By Lemma 8, we have Ψ∘kμ−≤Ψ∧μ−. Hence, Ψ∘kμ−=, Ψ∧μ−.
Conversely, supposeΨ∘kμ−=Ψ∧μ− for each ∈,∈∨q‐k-BFIRΨ=R,Ψn,Ψp, also for each ∈,∈∨q‐k–BFILμ=R,μn,μp of R. Suppose G and H are right k-ideal and left k-ideal of R, respectively. The lower components of BF characteristic function CG−=R,CnG−,CpG− is an ∈,∈∨q–k–BFIR and CH−=R,CnH−,CpH− is an ∈,∈∨q-k–BFIL of R. From our supposition, CG∧CH−=CG∘kCH−. This indicates that CG∩H−=CGH¯− implies G∩H=GH¯. Thus, by Lemma 1, R is k-regular.Theorem 5.
R is k-regular iff Ψ∧μ∧ν−≤Ψ∘kμ∘kν− for every ∈,∈∨q-k-BFIRΨ=R,Ψn,Ψp, for every ∈,∈∨q-k-BFbI μ=R,μn,μp and for all ∈,∈∨q-k-BFILν=R,νn,νp of R.Proof.
AssumeR is k-regular semiring then for all u∈R, there are elements c,c′∈R such as u+ucu=uc′u. Moreover, we have the following: (9)Ψn∘kμn∘kνn−u=Ψn∘kμn∘kνnu∨−0.5=∧u+∑i=1naib=∑j=1maj′bj′∨i=1nΨn∘kμnai∨∨i=1nνnbi∨∨j=1mΨn∘kμnaj′∨∨j=1mνnbj′∨−0.5≤Ψn∘kμnu∨νncu∨Ψn∘kμnu∨νnc′u∨−0.5=Ψn∘kμnu∨νncu∨νnc′u∨−0.5=∧u+∑i=1naib=∑j=1maj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′∨νncu∨νnc′u∨−0.5≤Ψnuc∨μnu∨Ψnuc′∨μnu∨νncu∨νnc′u∨−0.5≤Ψnu∨−0.5∨Ψnu∨−0.5∨μnu∨νnu∨−0.5∨νnu∨−0.5∨−0.5=Ψnu∨μnu∨νnu∨−0.5=Ψn∨μn∨νn−u
Similarly,Ψp∘kμp∘kνp−u≥Ψp∧μp∧νp−u. Consequently, Ψ∧μ∧ν−≤Ψ∘kμ∘kν−.
Conversely, letΨ=R,Ψn,Ψp is an ∈,∈∨q‐k‐BFIR and ν=R,νn,νp is an ∈,∈∨q‐k‐BFIL of R.
Then,Ψp∧νp−=Ψp∧R⌣p∧νp−≤Ψp∘kR⌣p∘kνp−=Ψp∘kνp− implies Ψp∧νp−≤Ψp∘kνp− but Ψp∧νp−≥Ψp∘kνp− by Lemma 8. Hence, Ψp∧νp−=Ψp∘kνp−. Similarly, Ψn∧νn−=Ψn∘kνn−. Hence, R is k-regular semiring by Theorem 4.Theorem 6.
R is k-regular if for each ∈,∈∨q-k-BFbI Ψ=R,Ψn,Ψp of R, we have Ψ−≤Ψ∘kR⌣∘kΨ−.Proof.
Straightforward.Theorem 7.
R is k-regular if Ψ∧μ−≤Ψ∘kμ∘kΨ− to each ∈,∈∨q-k-BFbI Ψ=R,Ψn,Ψp, and for every ∈,∈∨q-k-BFI μ=R,μn,μp of R.Proof.
AssumeR is k-regular, also to each u∈R, there are elements c,c′∈R such that u+ucu=uc′u. Thus, we have as follows:(10)Ψn∘kμn∘kΨn−u=Ψn∘kμn∘kΨnu∨−0.5=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨn∘kμnai∨∨i=1nΨnbi∨∨j=1mΨn∘kμnaj′∨∨j=1mΨnbj′∨−0.5≤Ψn∘kμnuc∨Ψnu∨Ψn∘kμnuc′∨Ψnu∨−0.5=Ψn∘kμnuc∨Ψnu∨Ψn∘kμnuc′∨−0.5=∧uc+∑i=1ndifi=∑j=1mdj′fj′∨i=1nΨndi∨∨i=1nμnfi∨∨j=1mΨndj′∨∨j=1mμnfj′∨Ψnu∨∧uc′+∑i=1ngihi=∑j=1mgj′hj′∨i=1nΨngi∨∨i=1nμnhi∨∨j=1mΨngj′∨∨j=1mμnhj′∨−0.5,
Sinceu+ucu=uc′u, uc+uceuc=uc′euc and uc′+uce′uc′=uc′e′uc′, so(11)≤Ψnu∨μnceuc∨Ψnu∨μnce′uc∨Ψnu∨Ψnu∨μnc′euc′∨Ψnu∨μnc′e′uc′∨−0.5≤Ψnu∨μnu∨−0.5∨−0.5=Ψnu∨μnu∨−0.5=Ψn∨μn−u.
Similarly,Ψp∘kμp∘kΨp−u≥Ψp∧μp−u. Hence, Ψ∧μ−≤Ψ∘kμ∘kΨ−.
Conversely, letΨ∧μ−≤Ψ∘kμ∘kΨ− for each ∈,∈∨q‐k‐BFbI Ψ=R,Ψn,Ψp and for each ∈,∈∨q‐k‐BFI μ=R,μn,μp of R. Let u∈R, then we have the following:(12)Ψn−u=Ψnu∨−0.5=Ψn∨R⌣nu∨−0.5=Ψn∨R⌣n−u≥Ψn∘kR⌣n∘kΨnuΨp−u=Ψpu∧0.5=Ψp∧R⌣pu∧0.5=Ψp∧R⌣p−u≤Ψp∘kR⌣p∘kΨp−u,which implies, Ψ−≤Ψ∘kR⌣∘kΨ−. So, by Theorem 6, R is k‐regular.Theorem 8.
For a semiringR, the given conditions are equivalent.(i)
R is k-regular semiring,(ii)
Ψ∧μ−≤Ψ∘kμ− for every ∈,∈∨q-k-BFbI Ψ=R,Ψn,Ψp and for every ∈,∈∨q-k-BFILμ=R,μn,μp of R.(iii)
Ψ∧μ−≤Ψ∘kμ− for each ∈,∈∨q-k-BFIRΨ=R,Ψn,Ψp, and for all ∈,∈∨q-k-BFbI μ=R,μn,μp of R.Proof.
(i)⟶ (ii)
SupposeR is a k-regular, so for u∈R, there are elements c,c′∈R such as u+ucu=uc′u. Then, Ψn∘kμn−u=Ψn∘kμnu∨−0.5(13)=∧u+∑i=1naibi=∑j=1maj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′∨−0.5≤Ψnu∨μncu∨Ψnu∨μnc′u∨−0.5≤Ψnu∨μnu∨−0.5∨Ψnu∨μnu∨−0.5∨−0.5=Ψnu∨μnu∨−0.5=Ψn∨μn−u.
Similarly,Ψp∘kμp−u≥Ψp∧μp−u. Consequently, Ψ∧μ−≤Ψ∘kμ−
(ii)⟶ (i)
SupposeΨ=R,Ψn,Ψp is an ∈,∈∨q‐k‐BFIR and μ=R,μn,μp is an ∈,∈∨q‐k‐BFIL of R. Since each ∈,∈∨q‐k‐BFIR is an ∈,∈∨q‐k‐BFbI of R. Thus by (ii), we get Ψ∧μ−≤Ψ∘kμ− but by Lemma 8, Ψ∘kμ−≤Ψ∧μ−. Hence, Ψ∧μ−=Ψ∘kμ− and by Theorem 4, R is k‐regular semiring.
Similarly, (i)⟶ (iii) and (iii) ⟶ (i) can be proved.Theorem 9.
For a semiringR, the given statements are equivalent:(i)
R is k-intra-regular.(ii)
Ψ∧μ−≤Ψ∘kμ− for every ∈,∈∨q-k–BFILΨ=R,Ψn,Ψp and ∈,∈∨q-k–BFIRμ=R,μn,μp of R.Proof.
SupposeR is a k-intraregular, so for each u∈R, there are elements ai,ai′,bj,bj′∈R such as u+∑i=1naiu2bi=∑j=1maj′u2bj′. That is, u+∑i=1naiuubi=∑j=1maj′uubj′
Then,Ψn∘kμn−u=Ψn∘kμnu∨−0.5(14)=∧u+∑i=1naibi=∑jmaj′bj′∨i=1nΨnai∨∨i=1nμnbi∨∨j=1mΨnaj′∨∨j=1mμnbj′∨−0.5≤Ψnaiu∨μnubi∨Ψnaj′u∨μnubj′∨−0.5≤Ψnu∨−0.5∨μnu∨−0.5∨Ψnu∨−0.5∨μnu∨−0.5∨−0.5=Ψnu∨μnu∨−0.5=Ψn∨μn−u.
Similarly,Ψp∘kμp−u≥Ψp∧μp−u . Consequently, Ψ∧μ−≤Ψ∘kμ−.
Conversely, LetG and H are left and right k-ideals of R, respectively. The lower components of the BF characteristic function CG−=R,CnG−,CpG− is an ∈,∈∨q-k–BFIR and CH−=R,CnH−,CpH− is an ∈,∈∨q-k–BFIL of R. From our supposition, CG∧CH−≤CG∘kCH−. This indicates that CG∩H−≤CGH¯− implies G∩H⊆GH¯. Thus R is k -intraregular by Lemma 3.


## 6. Comparative Study and Discussion

In [33], Shabir et al. characterized regular and intraregular semirings with the help of α,β–BFIs. We established the results of [33] in the framework of α,β–k–BFIs. We have characterized k-regular and k-intraregular semiring by using α,β-k-BFIs. As k-ideal is a more restricted form of the ideal in semiring, so our enlargement has expanded in the applications than the access deliberated in [33].


## 7. Conclusion

BFS is an important theory of mathematics to tackle the fuzziness in positive and negative features of an object. In this paper, we have studiedα,β-k-BFIs. Basic definitions, concepts, operations, and related properties with respect to α,β-k-BFIs and α,β-k-BFSSs are purposed. Many examples are illuminated to show the importance of k-ideals in semiring. Generally, we have proved with an example that if a BF subset Ψ of R is an ∈,∈∨q-k-BFSS of R, then it satisfies three particular conditions, but the reverse may not hold true. In addition, we have studied the lower and upper parts of ∈,∈∨q-k-BFIs of semiring. Also, we have characterized k-regular and k-intraregular semiring in terms of ∈,∈∨q-k-BFIs of semiring.A bipolar fuzzy set has membership degree ranges from −1 to 1, but we often face such critical situations in real life which cannot be handled by bipolar fuzzy sets due to their membership degree [−1, 1]. To control these critical situations, Pythagorean fuzzy sets are more useful because of sum of their membership degree and non-membership degree which can be greater than 1.In future, we will extend this work for hyperstructures, LA-semigroups, near-rings, etc. In addition, we will study its real life applications in medical science, computer science, management science, and many other fields.



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