# A New Approach to Study h-Hemiregular Hemirings in terms of Bipolar Fuzzy h-Ideals

Authors: Shahida Bashir; Ahmad N. Al-Kenani; Rabia Mazhar; Zunaira Pervaiz
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper provides a generalized form of ideals, that is, h-ideals of hemirings with the combination of a bipolar fuzzy set (BFS). The BFS is an extension of the fuzzy set (FS), which deals with complex and vague problems in both positive and negative aspects. The basic purpose of this paper is to introduce the idea ofα,β−bipolar fuzzy h-subhemirings (h-BFSHs), α,β−bipolar fuzzy h-ideals (h-BFIs), and α,β−bipolar fuzzy h-bi-ideals (h-BFbIs) in hemirings by applying the definitions of belongingness ∈ and quasicoincidence q of the bipolar fuzzy point. We will also focus on upper and lower parts of the h-product of bipolar fuzzy subsets (BFSSs) of hemirings. In the end, we have characterized the h-hemiregular and h-intrahemiregular hemirings in terms of the ∈,∈∨q−h-BFIs and ∈,∈∨q−h-BFbIs.


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## Body

## 1. Introduction and Motivation

In 1994, Zhang [1] introduced bipolar fuzzy set theory which is an inflation of fuzzy set theory. Bipolarity is an important idea that is mostly used in our daily life. In a lot of disciplines such as decision making, algebraic structures, graph theory, and medical science, bipolar valued fuzzy sets have become a significant research work. In real life, it is noticed that people may have a different response at a time for the same qualities of an item or a plan. One may have a positive response, and the other one may have a negative response; for example, $100 is a big amount for a needy person, but at the same time, this amount may have less value for a rich man. Similarly, sweetness and sourness of a food, effects and side effects of medicines, good and bad human behavior, happiness and sadness, thin and thick fluid, and honesty and dishonesty all are two-sided aspects of an object or situation. See [2–10] for examples and results which are relevant to bipolar fuzzy sets.In 1965, Zadeh [11] introduced the concept of fuzzy set theory which deals with the uncertain and complex problems in decision-making theory, medical science, engineering, automata theory, and graph theory [12–16]. The right place of entries to fuzzy set is indicated by a membership degree. In [0, 1] interval, perimeter point 0 shows no fuzzy set acceptability and 1 shows the fuzzy set acceptance. Also, (0, 1) defines the fuzzy collection to be partially belonging. If the membership degree is any property, then 1 describes that the element satisfies the property and 0 describes that the element does not satisfy the property. Interval (0, 1) shows the midway condition. But, there was a difficulty to deliberate the irrelevancy of data to the fuzzy set. The FS is extended to BFS to tackle such situations.In 1934, Vandiver [17] firstly familiarized the theory of semiring. In 1935, Von Neumann introduced the idea of regularity in rings and showed that for any nonempty set R, if the semigroup (R, ·) is regular, then the ring (R, +, ·) is also regular [18]. In 1951, Bourne showed if for all r ∈ R there exist x, y ∈ R such that r + rxr = ryr, then semiring (R, +, ·) is also regular [19]. Hemirings (semirings with zero and commutative addition) are studied in the theory of automata and formal languages [20–22]. Algebraic patterns are very important in mathematics. Hemiring is also a useful algebraic structure. It is very useful in functional analysis, physics, computation, coding, topological space, automata theory, formal language theory, mathematical modelling, and graph theory.In the structure theory of semirings, ideals play a vital role [23]. Henriksen gave in [24] a restricted class of ideals in semirings, which is k-ideals. Another more restricted class of ideals which is h-ideals has been given in hemirings by Iizuka [25]. However, in an additively commutative semiring, ideals of a semiring coincide with “ideals” of a ring, provided that the semiring is a hemiring [26, 27]. For more applications of h-ideals, see [28, 29].Ahsan et al. [30] presented applications of fuzzy semirings in automata theory. Since the bipolar fuzzy theory is a development of fuzzy theory, one may expect that bipolar fuzzy semirings will be useful in studying bipolar fuzzy automata theory and bipolar fuzzy languages.


### 1.1. Related Works

Zadeh’s fuzzy set is a much innovative, crucial, and useful set due to its significance in multiple research dimensions. Fuzzy set addresses the ill-defined situations by which we are often encountered. From these ill-defined situations, we can evaluate results by using a degree of membership of the fuzzy set but the bipolar fuzzy set is a much better set to manage uncertainty, vagueness, and impreciseness than the fuzzy set. A bipolar fuzzy set is much valuable due to its degree of membership [−1,1]. Latorre [31] discussed the properties of h-ideals of hemirings and presented some suitable results with respect to hemirings. Zhan and Dudek [28] proposed a model of fuzzy h-ideals with their useful properties. They presented some algebraic properties of prime fuzzy h-hemiregular hemirings. Kumaran et al. [32] discussed some basic properties of α,β-level subsets of bipolar valued fuzzy subsemiring of a hemiring and their related results. We extended this useful model and introduced h-hemiregular hemiring in terms of bipolar fuzzy h-ideals.


### 1.2. Historical Background

The bipolar fuzzy set describes the main idea which lies in the existence of bipolarity (positivity and negativity). In fact, human decision-making consists of double sides on the positive and negative aspects, for example, competition and cooperation, effect and side effect, and hostility and friendship. In traditional Chinese medicine, “Yin” is a negative side of a system and “Yang” is a positive side of a system. The equilibrium and coexistence of the two sides are keys for prosperity and stability of a social system [33].


### 1.3. Motivation

Bipolarity fuzzy sets have potential impacts on many fields, including information science, neural science, computer science, artificial intelligence, decision science, economics, cognitive science, and medical science [33]. Recently, the bipolarity fuzzy set has been studied in terms of a hemiring a bit increasingly and a bit enthusiastically. That is why we have been encouraged and motivated to introduce and study h-hemiregular hemiring in terms of a bipolar fuzzy set.


### 1.4. 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 another one may have a positive response at the same time for the same characteristic of an object. In this regard, bipolarity is a very useful concept that is commonly used in our real-world problems. Recently, Shabir et al. presented∈,∈∨q-fuzzy h-subhemirings and their related properties [34]. Bhakat and Das [35] firstly introduced the idea of α,β−fuzzy subgroups. Dudek et al. [36] worked on α,β−fuzzy ideals of hemirings. The idea of finite state machine on bipolar fuzzy theory is given by Jun and Kavikumar in [2]. Lee [37] introduced bipolar valued fuzzy ideals. Ibrar et al. [38] used α,β−bipolar fuzzy generalized bi-ideal for characterizations of regular ordered semigroups. In 2019, Shabir et al. [39] used α,β−bipolar fuzzy ideals and α,β−bipolar fuzzy bi-ideals for the characterizations of the regular and intraregular semiring. Recently, Anjum et al. [27] studied ordered h-ideals in regular semiring. Here, we have extended the study in [39] for α,β−h-BFSHs and α,β−h-BFIs of hemirings.


### 1.5. Organization of the Paper

In Section1, we have familiarized ourselves with the background of bipolar fuzzy hemiring and its characterizations towards regular and intraregular hemiring. Section 2 describes the literature review of h-BFSHs, h-BFIs, and some new important definitions which is used as the basic concept for the major works. In Section 3, the concepts of α,β−h-BFSHs and α,β−h-BFIs of hemirings are discussed. In Section 4, we have worked on the upper and lower parts of BFSH by h-product. In Section 5, the description of theorems of h-hemi-regular and h-intrahemiregular hemirings in terms of the ∈,∈∨q−h-BFIs and ∈,∈∨q−h-BFbIs is given. In Section 6, the comparative study is given, and the last section consists of the conclusions and future plans.The list of acronyms used here is given in Table1.Table 1
List of acronyms.
AcronymsRepresentationFSFuzzy setBFSBipolar fuzzy setBFSSBipolar fuzzy subseth-BFSHBipolar fuzzy h-subhemiringh-BFIBipolar fuzzy h-idealh-BFILBipolar fuzzy left h-idealh-BFIRBipolar fuzzy right h-idealh-BFbIBipolar fuzzy h-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. Fuzzy set addresses the ill-defined situations by which we are often encountered. From these ill-defined situations, we can evaluate results by using a degree of membership of the fuzzy set but the bipolar fuzzy set is a much better set to manage uncertainty, vagueness, and impreciseness than the fuzzy set. A bipolar fuzzy set is much valuable due to its degree of membership [−1,1]. Latorre [31] discussed the properties of h-ideals of hemirings and presented some suitable results with respect to hemirings. Zhan and Dudek [28] proposed a model of fuzzy h-ideals with their useful properties. They presented some algebraic properties of prime fuzzy h-hemiregular hemirings. Kumaran et al. [32] discussed some basic properties of α,β-level subsets of bipolar valued fuzzy subsemiring of a hemiring and their related results. We extended this useful model and introduced h-hemiregular hemiring in terms of bipolar fuzzy h-ideals.


## 1.2. Historical Background

The bipolar fuzzy set describes the main idea which lies in the existence of bipolarity (positivity and negativity). In fact, human decision-making consists of double sides on the positive and negative aspects, for example, competition and cooperation, effect and side effect, and hostility and friendship. In traditional Chinese medicine, “Yin” is a negative side of a system and “Yang” is a positive side of a system. The equilibrium and coexistence of the two sides are keys for prosperity and stability of a social system [33].


## 1.3. Motivation

Bipolarity fuzzy sets have potential impacts on many fields, including information science, neural science, computer science, artificial intelligence, decision science, economics, cognitive science, and medical science [33]. Recently, the bipolarity fuzzy set has been studied in terms of a hemiring a bit increasingly and a bit enthusiastically. That is why we have been encouraged and motivated to introduce and study h-hemiregular hemiring in terms of a bipolar fuzzy set.


## 1.4. 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 another one may have a positive response at the same time for the same characteristic of an object. In this regard, bipolarity is a very useful concept that is commonly used in our real-world problems. Recently, Shabir et al. presented∈,∈∨q-fuzzy h-subhemirings and their related properties [34]. Bhakat and Das [35] firstly introduced the idea of α,β−fuzzy subgroups. Dudek et al. [36] worked on α,β−fuzzy ideals of hemirings. The idea of finite state machine on bipolar fuzzy theory is given by Jun and Kavikumar in [2]. Lee [37] introduced bipolar valued fuzzy ideals. Ibrar et al. [38] used α,β−bipolar fuzzy generalized bi-ideal for characterizations of regular ordered semigroups. In 2019, Shabir et al. [39] used α,β−bipolar fuzzy ideals and α,β−bipolar fuzzy bi-ideals for the characterizations of the regular and intraregular semiring. Recently, Anjum et al. [27] studied ordered h-ideals in regular semiring. Here, we have extended the study in [39] for α,β−h-BFSHs and α,β−h-BFIs of hemirings.


## 1.5. Organization of the Paper

In Section1, we have familiarized ourselves with the background of bipolar fuzzy hemiring and its characterizations towards regular and intraregular hemiring. Section 2 describes the literature review of h-BFSHs, h-BFIs, and some new important definitions which is used as the basic concept for the major works. In Section 3, the concepts of α,β−h-BFSHs and α,β−h-BFIs of hemirings are discussed. In Section 4, we have worked on the upper and lower parts of BFSH by h-product. In Section 5, the description of theorems of h-hemi-regular and h-intrahemiregular hemirings in terms of the ∈,∈∨q−h-BFIs and ∈,∈∨q−h-BFbIs is given. In Section 6, the comparative study is given, and the last section consists of the conclusions and future plans.The list of acronyms used here is given in Table1.Table 1
List of acronyms.
AcronymsRepresentationFSFuzzy setBFSBipolar fuzzy setBFSSBipolar fuzzy subseth-BFSHBipolar fuzzy h-subhemiringh-BFIBipolar fuzzy h-idealh-BFILBipolar fuzzy left h-idealh-BFIRBipolar fuzzy right h-idealh-BFbIBipolar fuzzy h-bi-idealIffIf and only if


## 2. Preliminaries

A semiring is an algebraic systemM,+,⋅ consisting of a nonempty set M together with two binary operations addition and multiplication such that M,+ and M,⋅ are semigroups satisfying for all u,v,w∈M, the following distributive laws uv+w=uv+uw and u+vw=uw+vw. By zero, we mean an element 0∈M such that 0⋅u=u⋅0=0 and 0+u=u+0=u for all u∈M. A semiring with zero and a commutative semigroup M,+ is known as hemiring.A nonempty subsetN of a semiring M is called a subhemiring of M if N is a hemiring under the induced operations of addition and multiplication of M. A nonempty subset “N” of a hemiring M is called a left (right) ideal of M if N is closed under “+” and mr∈Nrm∈N for all m∈M and r∈N, and N is called a two-sided ideal or simply an ideal of M if it is both a left and a right ideal of M.A hemiringM is said to be h-hemiregular if, for each x∈M, there exist a1,a2,z∈M such that x+xa1x+z=xa2x+z, and M is said to be h-intra-hemiregular if, for each x∈M, there exist ai,aj′,bi,bj′,z∈R such that x+∑i=1maix2bi+z=∑j=1naj′x2bj′+z [29]. Throughout the paper, M is hemiring unless otherwise identified.Lemma 1 (see [29]).
IfM is h-intrahemiregular iff for any left h-ideal L and right h-ideal N, L∩N⊆LN¯.Lemma 2 (see [29]).
IfM is h-hemiregular iff for any right h-ideal L and any left h-ideal N, we have LN¯=L∩N.Definition 1 (see [1]).
Letη=M,ηn,ηp and ζ=M,ζn,ζp be two BFSSs of M. Then, h-product η∘hζ=M,ηn∘hζn,ηp∘hζp is defined as follows:(1)ηn∘hζnu=∧u+∑i=1mcidi+v=∑j=1ncj′dj′+vηnci∨ηncj′∨ζndi∨ζndj′,ηp∘hζpu=∨u+∑i=1mcidi+v=∑j=1ncj′dj′+vηpci∧ηpcj′∧ζpdi∧ζpdj′,ηn∘hζnu=0,ηp∘hζpu=0ifuis not representable asu+∑i=1mcidi+v=∑j=1ncj′dj′+v.Definition 2.
(see [1]). If H is an h-subset of M. Then, the bipolar fuzzy characteristic function on H is denoted by χH=M;χnH,χpH and is defined by χnHu=−1ifu∈H0ifu∉H, χpHu=1ifu∈H0ifu∉H for all u∈M. If H=M, then we have BFSS M=M;Mn,Mp defined as M˜nu=−1 and M˜pu=1 for all u∈M.Definition 3.
(see [1]). A BFSSη=M,ηn,ηp of M is called h-BFSH of M if it satisfies the following:(a)
ηn0≤ηnu and ηp0≥ηpu(b)
ηnu+v≤maxηnu,ηnv and ηpu+v≥minηpu,ηpv(c)
ηnuv≤maxηnu,ηnv and ηpuv≥minηpu,ηpv(d)
Ifu+c+v=d+v, then ηnu≤maxηnc,ηnd and ηpu≥minηpc,ηpd for all c,d,u,v∈MDefinition 4.
A BFSSη=M,ηn,ηp of M is said to be an h-BFIL (resp., h-BFIR) of M, if it satisfies the following:(a)
ηnu+v≤maxηnu,ηnv and ηpu+v≥minηpu,ηpv(b)
ηnuv≤ηnv and ηpuv≥ηpv (resp., ηnuv≤ηnu and ηpuv≥ηpu)(c)
Ifu+c+v=d+v, then ηnu≤maxηnc,ηnd and ηpu≥minηpc,ηpd for all c,d,u,v∈MExample 1.
Consider a BFSSη of a hemiring M=0,1,2,3,… as ηnu=0 ifu=0−0.5otherwise and ηpu=0 ifu=00.5otherwise.
Then,η is a BFI but not a h-BFI of M because if we take x=0,a=1,b=1andz=5, then for 0+1+5=1+5,ηn0=0≥−0.5=maxηn1,ηn5 and ηp0=0≤0.5=minηp1,ηp5.Example 2.
Consider a hemiringM=0,a,b,c under the operations as in Tables 2 and 3.
Define a BFSSη=M,ηn,ηp on M as shown in Table 4.
Then, it is easy to check thatη is a h-BFI of M.Table 2
Binary addition.
+0abc00abcaaabcbbbbcccccbTable 3
Binary multiplication.
.0abc00000a0bbbb0bbbc0bbbTable 4
Values of BFSS on M.
M0abcηp0.50.50.50.2ηn−0.5−0.5−0.5−0.1Definition 5.
A BFSSη=M,ηn,ηp of M is called an h-BFbI of M, if is satisfies the following:(a)
ηn0≤ηnu and ηp0≥ηpu(b)
ηnu+v≤maxηnu,ηnv and ηpu+v≥minηpu,ηpv(c)
ηnuv≤maxηnu,ηnv and ηpuv≥minηpu,ηpv(d)
ηnuvw≤ max ηnu,ηnw and ηpuvw≥ min ηpu,ηpw(e)
Ifu+c+v=d+v, then ηnu≤maxηnc,ηnd and ηpu≥minηpc,ηpd for all c,d,u,v,w∈MLemma 3.
If H is a left h-ideal (resp., right) ofM, then χH=M;χnH,χpH is an h-BFIL (resp., h-BFIR) of M.Proof: .
We have to prove the following three inequalities for left h-ideal:(a)
χnHu+v≤maxχnHu,χnHv and χpHu+v≥minχpHu,χpHv(b)
χnHuv≤χnHv and χpHuv≥χpHv(c)
Ifu+c+v=d+v, then χnHu≤maxχnHc,χnHd and χpHu≥minχpHc,χpHd for all c,d,u,v∈H
For the proof of parts (a) and (b), see [2]. Here, we just prove part (c). For this, we discuss cases for all c,d,u,v∈H,u+c+v=d+v.Case 1.
Ifu,c,d∈H, then χnHu=−1=χnHv=χnHc=χnHd and χpHu=1=χpHv=χpHc=χpHd.
Then,χnHu=−1≤maxχnHc,χnHd and χpHu=1≥minχpHc,χpHd.Case 2.
Ifu,c,d∉H, then χnHu=0=χnHv=χnHc=χnHd and χpHu=0=χpHv=χpHc=χpHd.
Then,χnHu=0≤maxχnHc,χnHd and χpHu=0≥minχpHc,χpHd.Case 3.
If one ofu,c,andd does not belong to H, say c∉H, then χnHc=0=χpHc,χnHu=−1=χnHv=χnHd, and χpHu=1=χpHv=χpHd. This implies that χnHu=−1≤0=maxχnHc,χnHd and χpHu=1≥0=minχpHc,χpHd.Case 4.
If any two ofu,c,andd do not belong to H, say u,c∉H, then χnHu=0=χnHc, χpHu=0=χpHc, χnHd=−1, and χpHd=1. This implies that χnHu=0≤maxχnHc,χnHd and χpHu=0≥minχpHc,χpHd.
Hence,χH=M;χnH,χpH is an h-BFIL of M.


## 3.α,β−Bipolar Fuzzy h-Ideals

In this section, we will introduce the concept ofα,β−h-BFSHs, α,β−h-BFIs, and α,β−h-BFbIs of hemiring and properties of such kinds of ideals are discussed.Definition 6.
(see [2]). Let c,d∈−1,0×0,1; then, t/c,d is called bipolar fuzzy point in M with BFSS η=M,ηn,ηp of M if ηpu=difu=t0ifu≠t and ηnu=cifu=t0ifu≠t for all u,t∈M.Definition 7.
(see [2]). A bipolar fuzzy point t/c,d belongs to (resp., quasicoincident to) a BFSS η=M,ηn,ηp as follows:(a)
t/c,d∈η if ηnt≤c and ηpt≥d(b)
t/c,dqη if ηnt+c<−1 and ηpt+d>1(c)
t/c,d∈∨qη if t/c,d∈η or t/c,dqη(d)
t/c,d∈∧qη if t/c,d∈η and t/c,dqη
In this paper, we considerα,β∈∈,q,∈∨q,∈∧q and α≠∈∧q. Consider a BFSS η=R,ηn,ηp such that ηnt≥−0.5 and ηpt≥0.5 for all t∈R and t/c,d∈∧qη. Then, ηnt≤c, ηpu≥d, ηnt+c<−1, and ηpu+d>1. It follows that −1>ηnt+c≥ηnt+ηnt=2ηnt which implies that ηnt≤−0.5, thus a contradiction. So, t/c,d∈∧q¯η, and hence, α≠∈∧q.Definition 8.
A BFSSη=M,ηn,ηp of M is said to be an α,β−h-BFSH of M if for all t,u∈M and c1,d1,c2,d2∈−1,0×0,1, it satisfies the following:(a)
t/c1,d1αη and u/c2,d2αη⟶t+u/c1∨c2,d1∧d2βη(b)
t/c1,d1αη and u/c2,d2αη⟶tu/c1∨c2,d1∧d2βη(c)
Ifc,d,t,u∈M such that t+c+u=d+u, then c/c1,d1αη and d/c2,d2αη⟶t/c1∨c2,d1∧d2βη; here, α≠∈∧qDefinition 9.
A BFSSη=M,ηn,ηp of M is known as an α,β−h-BFIL (resp., α,β−h-BFIR) of M if for all t,u∈M and c1,d1,c2,d2∈−1,0×0,1, it satisfies the following:(a)
t/c1,d1αη and u/c2,d2αη⟶t+u/c1∨c2,d1∧d2βη(b)
t/c1,d1αη and u∈M⟶ut/c1,d1βη (rep. tu/c1,d1βη)(c)
Ifc,d,t,u∈M such that t+c+u=d+u, then c/c1,d1αη and d/c2,d2αη⟶t/c1∨c2,d1∧d2βη; here, α≠∈∧qDefinition 10.
A BFSSη=M,ηn,ηp of M is known as an α,β−h-BFbI of M if for all t,u,y∈M and c1,d1,c2,d2∈−1,0×0,1, it satisfies the following:(a)
t/c1,d1αη and u/c2,d2αη⟶t+u/c1∨c2,d1∧d2βη(b)
t/c1,d1αη and u/c2,d2αη⟶tu/c1∨c2,d1∧d2βη(c)
t/c1,d1αη and u/c2,d2αη⟶tyu/c1∨c2,d1∧d2βη(d)
Ifc,d,t,u∈M such that t+c+u=d+u, then c/c1,d1αη and d/c2,d2αη⟶t/c1∨c2,d1∧d2βη; here, α≠∈∧q
Throughout this paper, our main focus will be on∈,∈∨q−h-BFSHs, ∈,∈∨q−h-BFIs, and ∈,∈∨q−h-BFBIs of M.Lemma 4.
For a BFSSη=M,ηn,ηp of M, if c,d,t,u∈M such that t+c+u=d+u, then c/c1,d1αη and d/c2,d2αη⟶t/c1∨c2,d1∧d2βη; then, ηnt≤maxηnc,ηnd,−0.5 and ηpt≥minηpc,ηpd,0.5.Proof: .
Letc,d,t,u∈M; then, we have the following four cases:(i)
maxηnc,ηnd≤−0.5 and minηpc,ηpd≥0.5(ii)
maxηnc,ηnd≤−0.5 and minηpc,ηpd<0.5(iii)
maxηnc,ηnd>−0.5 and minηpc,ηpd≥0.5(iv)
maxηnc,ηnd>−0.5 and minηpc,ηpd<0.5
Contrarily assume forc,d∈M,ηnt>maxηnc,ηnd,−0.5 or ηpt<minηpc,ηpd,0.5.For case-(i):ηnt>−0.5 or ηpt<0.5; this implies t/−0.5,0.5∈¯η. Now, consider maxηnc,ηnd≤−0.5 and minηpc,ηpd≥0.5 and then c/−0.5,0.5∈η and d/−0.5,0.5∈η but for t∈M,t/−0.5,0.5∈¯η. In the same way, t/−0.5,0.5q¯η because ηnt−0.5>−1 or ηpt+0.5<1. So, t/−0.5,0.5∈∨q¯η which is a contradiction.For case-(ii):ηnt>−0.5 or ηpt<minηpc,ηpd; so for s∈0,0.5, ηpt<s=minηpc,ηpd implies t/−0.5,s∈¯η. Now, consider again maxηnc,ηnd≤−0.5 and minηpc,ηpd=s and then c/−0.5,s∈η and d/−0.5,s∈η but for t∈M,t/−0.5,s∈¯η. In the same way, t/−0.5,sq¯η because ηnt−0.5>−1 or ηpt+s<1. So, t/−0.5,s∈∨q¯η, and it contradicts our supposition.For case-(iii):ηnt>maxηnc,ηnd or ηpt<0.5. So, for r∈−0.5,0, ηnt>r=maxηnc,ηnd implies t/r,0.5∈¯η. Also, maxηnc,ηnd=r and minηpc,ηpd≥0.5 and then we obtain c/r,0.5∈η and d/r,0.5∈η but for t∈M,t/r,0.5∈¯η. Similarly, t/r,0.5q¯η because ηnt+r>−1 or ηpt+0.5<1. Therefore, t/r,0.5∈∨q¯η, which is a contradiction.For case-(iv):ηnt>maxηnc,ηnd or ηpt<minηpc,ηpd. For r∈−0.5,0 and s∈0,0.5, ηnt>r=maxηnc,ηnd and ηpt<s=minηpc,ηpd and then t/r,s∈¯η. Also, maxηnc,ηnd=r and minηpc,ηpd=s; then, we obtain c/r,s∈η and d/r,s∈η but for t∈M,t/r,s∈¯η. Similarly, t/r,sq¯η because ηnt+r>−1 or ηpt+s<1. Therefore, t/r,s∈∨q¯η, which is a contradiction.
Therefore,ηnt≤maxηnc,ηnd,−0.5 and ηpt≥minηpc,ηpd,0.5. The reverse of this theorem does not hold in general, shown by the following example.Example 3.
Consider a BFSSη in Example 2; for t+c+u=d+u,η satisfies ηnt≤maxηnc,ηnd,−0.5 and ηpt≥minηpc,ηpd,0.5. But if c/c1,d1∈η and d/c2,d2∈η, then t/c1∨c2,d1∧d2∈∨qη does not hold. Because for b+0+c=a+c, 0/−0.5,0.5∈η and a/−0.5,0.5∈η, but b/−0.5,0.5∈∨q¯η as ηpb=0.2<0.5 and ηnb=−0.1>−0.5. Also, ηpb+0.5=0.2+0.5<1 and ηnb+−0.5=−0.1+−0.5>−1 implies b/−0.5,0.5q¯η. So, b/−0.5,0.5∈∨q¯η.Theorem 1.
If a BFSSη=M,ηn,ηp of M is an ∈,∈∨q -h-BFSH of M, then it fulfills the following conditions for each c,d,t,u∈M:(a)
ηnc+d≤maxηnc,ηnd,−0.5 and ηpc+d≥minηpc,ηpd,0.5(b)
ηncd≤maxηnc,ηnd,−0.5 and ηpcd≥minηpc,ηpd,0.5(c)
Fort+c+u=d+u,ηnt≤maxηnc,ηnd,−0.5 and ηpt≥minηpc,ηpd,0.5Proof: .
Follows from Lemma4 and [2].Theorem 2.
If a BFSSη=M,ηn,ηp of M is an ∈,∈∨q -h-BFIL (resp., ∈,∈∨q -h-BFIR) of M, then it fulfills the following conditions for each c,d,t,u∈M:(a)
ηnc+d≤maxηnc,ηnd,−0.5 and ηpc+d≥minηpc,ηpd,0.5(b)
ηncd≤maxηnd,−0.5 and ηpcd≥minηpd,0.5 (resp., ηncd≤maxηnc,−0.5 and ηpcd≥minηpc,0.5)(c)
Fort+c+u=d+u,ηnt≤maxηnc,ηnd,−0.5 and ηpt≥minηpc,ηpd,0.5Proof: .
Follows from Lemma4 and [2].Theorem 3.
If a BFSSη=M,ηn,ηp of M is an ∈,∈∨q-h-BFbI of M, then it fulfills the following conditions for each c,d,t,u∈M:(a)
ηnc+d≤maxηnc,ηnd,−0.5 and ηpc+d≥minηpc,ηpd,0.5(b)
ηncd≤maxηnc,ηnd,−0.5 and ηpcd≥minηpc,ηpd,0.5(c)
ηncbd≤maxηnc,ηnd,−0.5 and ηpcbd≥minηpc,ηpd,0.5(d)
Fort+c+u=d+u,ηnt≤maxηnc,ηnd,−0.5 and ηpt≥minηpc,ηpd,0.5Proof: .
Follows from Lemma4 and [2].


## 4. Upper and Lower Parts of a Bipolar Fuzzy Set

In the following segment, we have offered the definitions and several stimulating outcomes regarding the upper parts and lower parts of h-BFSS.Definition 11.
(see [2]). If η  =  M,ηn,ηp is a bipolar fuzzy h-set in M, then the upper part η+=M,ηn+,ηp+ of η is written as ηn+t=ηnt∧−0.5 and ηp+t=ηpt∨0.5. The lower part of η is denoted by η−=M,ηn−,ηp− and is defined as ηn−t=ηnt∨−0.5 and ηp−t=ηpt∧0.5for allt∈M.Lemma 5.
Letη  =  M,ηn,ηp and ζ=M,ζn,ζp be two BFSHs in M. Then, the following equations hold:(a)
η∧ζ−=η−∧ζ−(b)
η∨ζ−=η−∨ζ−(c)
η∘hζ−=η−∘hζ−Proof: .
Straightforward.Lemma 6.
Letη  =  M,ηn,ηp and ζ=M,ζn,ζp be two BFSHs in M. Then, the following equations hold:(a)
η∧ζ+=η+∧ζ+(b)
η∨ζ+=η+∨ζ+(c)
η∘hζ+=η+∘hζ+Proof: .
Straightforward.Lemma 7.
For an∈,∈∨q−h-BFIRη  =  M,ηn,ηp and ∈,∈∨q−h-BFILζ  =  M,ζn,ζp, then we have η∘hζ−≤η∧ζ−.Proof: .
Lett∈M if η∘hζt=0, that is, ηn∘hζnt=0, and then ηn∘hζn−t=0∨−0.5=0≥ηn∨ζn−t and ηp∘hζpt=0 indicates ηp∘hζp−t=0∧0.5=0≤ηp∧ζp−t. Otherwise,(2)ηn∘hζn−t=ηn∘hζnt∨−0.5=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnai∨∨i=1mζnbi∨∨j=1nηncj∨∨j=1nζndj∨−0.5=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnai∨−0.5∨∨i=1mζnbi∨−0.5∨∨j=1nηncj∨−0.5∨∨j=1nζndj∨−0.5∨−0.5≥∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnaibi∨∨i=1mζnaibi∨∨j=1nηncjdj∨∨j=1nζncjdj∨−0.5=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnaibi∨−0.5∨∨i=1mζnaibi∨−0.5∨∨j=1nηncjdj∨−0.5∨∨j=1nζncjdj∨−0.5∨−0.5≥∧t+∑i=1maibi+v=∑j=1ncjdj+vηn∑i=1maibi∨ζn∑i=1maibi∨ηn∑j=1ncjdj∨ζn∑j=1ncjdj∨−0.5=ηn∨ζnt∨−0.5.
This impliesηn∘hζn−t≥ηn∨ζn−t.
Similarly,ηp∘hζp−t≤ηp∧ζp−t.
Hence,η∘hζ−≤η∧ζ−.Definition 12.
If H is an h-subset ofM, the upper part χH+=M;χnH+,χpH+ of the bipolar fuzzy characteristic function χH=M;χnH,χpH of H is defined by the following:(3)χnH+t=−1if t∈H,−0.5if t∉H,χpH+t=1if t∈H,0.5if t∉H.
Similarly, the lower partχH−=M;χnH−,χpH− of the bipolar fuzzy characteristic function χH=M;χnH,χpH of H is defined by the following:(4)χnH−t=−0.5if t∈H,0if t∉H,χpH−t=0.5if t∈H,0if t∉H.
For allt∈M, if H=M, then we have BFSS M=M;Mn,Mp defined as M˜nt=−1 and M˜pt=1 for all t∈M.Lemma 8.
LetE and F be two nonempty h-subsets of M. Then, the following results hold:(a)
χE∧χF−=χE∩F−(b)
χE∨χF−=χE∪F−(c)
χE∘hχF−=χEF¯−Proof: .
Proofs of (a) and (b) are straightforward. Here is just a proof of (c).
Supposet∈M and t∈EF¯. Then, χnEF¯t=−0.5 and χpEF¯t=0.5. For t∈EF¯,t+∑i=1maibi+v=∑j=1ncjdj+v where ai,cj∈E and bi,dj∈F.; then, we have χnEai=−1, χpEcj=1, χnFbi=−1, and χpFdj=1.
Consider the following:(5)χnE∘hχnF−t=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mχnEai∨∨i=1mχnFbi∨∨j=1nχnEcj∨∨j=1nχnFdj∨−0.5=−1∨−1∨−1∨−1∨−0.5=−0.5=χnEF¯−t.
Now,(6)χpE∘hχpF−t=∨t+∑i=1maibi+v=∑j=1ncjdj+v∧i=1mχnEai∧∧i=1mχnFbi∧∧j=1nχnEcj∧∧j=1nχnFdj∧0.5=1∧1∧1∧1∧0.5=0.5=χpEF¯−t.
Hence,χE∘hχF−=χEF¯−.
Ift∉EF¯, then χnEF¯t=0 and also χpEF¯t=0.
Then,χnE∘hχnF−t=χnE∘hχnFt∨−0.5=0∨−0.5=0=χnEF¯−t.
And,χpE∘hχpF−t=χpE∘hχpFt∧0.5=0∧0.5=0=χpEF¯−t.
That is,χE∘hχF−=χEF¯−.Lemma 9.
If H is left (resp., right) h-ideal ofM, then χH−=M;χnH−,χpH− is ∈,∈∨q−h-BFIL (resp., ∈,∈∨q−h-BFIR) of M.Proof: .
Straightforward.Lemma 10.
If H is h-bi-ideal ofM, then χH−=M;χnH−,χpH− is ∈,∈∨q−h-BFbI of M.Proof: .
Straightforward.


## 5. h-Hemiregular and h-Intrahemiregular Hemirings

In this section, by using the h-BFIs, theorems on hemiregular and intrahemiregular hemirings are presented.Theorem 4.
SupposeM is a hemiring, then the following subsequent conditions are identical:(a)
M is h-hemiregular(b)
η∧ζ−=η∘hζ− for all ∈,∈∨q−h-BFIRη=M,ηn,ηp and ∈,∈∨q−h-BFILζ=M,ζn,ζp of MProof: .
LetM be an h-hemi-regular hemiring. For each t∈M, there exist x1,x2,y∈M such that t+tx1t+y=tx2t+y; then,(7)ηn∘hζn−t=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnai∨∨i=1mζnbi∨∨j=1nηncj∨∨j=1nζndj∨−0.5≤ηntx1∨ζnt∨ηntx2∨ζnt∨−0.5≤ηnt∨−0.5∨ζnt∨ηnt∨−0.5∨ζnt∨−0.5=ηnt∨ζnt∨−0.5=ηn∨ζn−t.
In the same way,ηp∘hζp−t≥ηp∧ζp−t.
Thus,η∧ζ−≤η∘hζ−. But, we have η∘hζ−≤η∧ζ− by Lemma 7. Hence, η∧ζ−=η∘hζ−.
Conversely,η∧ζ−=η∘hζ− for all ∈,∈∨q−h-BFIRη=M,ηn,ηp and ∈,∈∨q−h-BFILζ=M,ζn,ζp of M. Let E and F be right h-ideal and left h-ideal of M correspondingly. Formerly, the lower parts of the bipolar fuzzy characteristic function χE−=M;χnE,χpE are an ∈,∈∨q−h-BFIR and χF−=M;χnF,χpF are an ∈,∈∨q−h-BFIL of M. Therefore, by our supposition, χE∧χF−=χE∘χF−. This indicates that χE∩F−=χEF¯ implies E∩F=EF¯. Therefore, by Lemma 2, M is h-hemiregular.Theorem 5.
For a hemiringM, the following subsequent conditions are identical:(a)
M is h-hemiregular(b)
η∧ζ∧υ−≤η∘hζ∘hυ− for each ∈,∈∨q−h-BFIRη=M,ηn,ηp, for all ∈,∈∨q−h-BFI ζ=M,ζn,ζp and for all ∈,∈∨q−h-BFILυ=M,υn,υp of MProof: .
Same as Theorem4.Theorem 6.
SupposeM is hemiring, then the following conditions are equivalent:(a)
M is h-hemiregular(b)
η∧ζ−≤η∘hζ− for all ∈,∈∨q−h-BFbI η=M,ηn,ηp and for all ∈,∈∨q−h-BFILζ=M,ζn,ζp of M(c)
η∧ζ−≤η∘hζ− for all ∈,∈∨q−h-BFIRη=M,ηn,ηp and for all ∈,∈∨q−h-BFbI ζ=M,ζn,ζp of MProof: .
(a)⟶(b): Let M be an h-hemiregular. For each t∈M, there exists x1,x2,y∈M such that t+tx1t+y=tx2t+y; then,(8)ηn∘hζn−t=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnai∨∨i=1mζnbi∨∨j=1nηncj∨∨j=1nζndj∨−0.5≤ηnt∨ζnx1t∨ηnt∨ζnx2t∨−0.5≤ηnt∨ζnt∨−0.5∨ηnt∨ζnt∨−0.5∨−0.5=ηnt∨ζnt∨−0.5=ηn∨ζn−t.
In the same way,ηp∘hζp−t≥ηp∧ζp−t. Thus, η∧ζ−≤η∘hζ−.
(b)⟶ (a): Let η=M,ηn,ηp be a ∈,∈∨q−h-BFIR and ζ=M,ζn,ζp be a ∈,∈∨q−h-BFIL of M. Since each ∈,∈∨q−h-BFIR is ∈,∈∨q−h-BFbI of M, by (a), we obtain η∧ζ−≤η∘hζ− but by Lemma 7, η∘hζ−≤η∧ζ−. Hence, η∧ζ−=η∘hζ−, and by Theorem 4, M is h-hemiregular. Similarly, we can prove (a) ⟶ (c) and (c) ⟶ (a).Theorem 7.
For a hemiringM, the following subsequent conditions are equivalent:(a)
M is h-intrahemiregular(b)
η∧ζ−≤η∘hζ− for all ∈,∈∨q−h-BFILη=M,ηn,ηp and for all ∈,∈∨q−h-BFIRζ=M,ζn,ζp of MProof: .
AssumeM is h-intrahemiregular hemiring. For every t∈M, there exists xi,zi,xj′,zj′,y∈M such that t+∑i=1nxit2zi+v=∑j=1mxj′t2zj′+v.(9)ηn∘hζn−t=∧t+∑i=1maibi+v=∑j=1ncjdj+v∨i=1mηnai∨∨i=1mζnbi∨∨j=1nηncj∨∨j=1nζndj∨−0.5≤ηnxit∨ζntzi∨ηnxjt∨ζntzj∨−0.5≤ηnt∨−0.5∨ζnt∨−0.5∨ηnt∨−0.5∨ζnt∨−0.5∨−0.5=ηnt∨ζnt∨−0.5=ηn∨ζn−t.
In the same way,ηp∘hζp−t≥ηp∧ζp−t.
Thus,η∧ζ−≤η∘hζ−.
Conversely, assumeE and F are left and right h-ideal of M, respectively. The lower parts of the bipolar fuzzy characteristic function of E and F, χE−=M,χnE−,χpE− and χF−=M,χnF−,χpF−, are ∈,∈∨q−h-BFIL and ∈,∈∨q−h-BFIR of M, respectively. Now, from our supposition, χE∧χF−≤χE∘hχF−. Also, χE∩F−=χE∧χF−≤χE∘hχF−=χEF¯− which suggests that E∩F⊆EF¯. From Lemma 1, M is h-intrahemiregular.


## 6. Comparative Study and Discussion

In [2], Shabir et al. used regular and intraregular semirings with the help of α,β−BFIs. We extended the work of [2] to hemirings with the help of α,β−h-BFIs. Semiring with zero and commutative addition is hemiring. We have characterized h-hemiregular and h-intrahemiregular hemirings by using α,β−h-BFIs. As h-ideal is a more restricted form of ideal, our extension is more applicable than the approach discussed in [2].


## 7. Conclusion

BFS is a dominant tool of mathematics to resolve the uncertainty in positive as well as a negative aspect of the data. In this paper, basic concepts, operations, and related properties with respect toα,β−h-BFIs are proposed. Generally, we have proved with an example that if a BFSS η of M is an ∈,∈∨q−h-BFI of M, then it satisfies three particular conditions, but the reverse may not hold. Also, we have studied the lower and upper parts of ∈,∈∨q−h-BFI of hemirings. We have characterized h-hemiregular and h-intrahemiregular hemirings by using α,β−h-BFIs.The 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 the sum of its membership degree and nonmembership degree which can be greater than 1.In the future, we will extend this work for hyperstructures, LA-semigroups, near-rings, etc. We will utilize this proposed bipolar fuzzy model to enhance these studies such as fuzzy stochastic data envelopment analysis model, intuitionistic fuzzy linear regression model, bounded linear programs with trapezoidal fuzzy numbers, interval-valued trapezoidal fuzzy number, fuzzy arc waves based on artificial bee colony algorithm, and fuzzy efficiency measures in data envelopment analysis. In addition, we will study its real-life applications in medical science, computer science, management science, and many other fields.



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