# Topological Structures of Lower and Upper Rough Subsets in a Hyperring

Authors: Nabilah Abughazalah; Naveed Yaqoob; Kiran Shahzadi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we study the connection between topological spaces, hyperrings (semi-hypergroups), and rough sets. We concentrate here on the topological parts of the lower and upper approximations of hyperideals in hyperrings and semi-hypergroups. We provide the conditions for the boundary of hyp-ideals of a hyp-ring to become the hyp-ideals of hyp-ring.


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## Body

## 1. Introduction

Algebraic hyp-structure (hyperstructure) represents a real extension of classical algebraic structure. Algebraic hyp-structures depend on hyperoperations and their properties. Sm-hyp-group (semi-hypergroup) was first introduced by French Mathematician Marty [1] in 1934. The sm-hyp-group concept is the generalization of sm-group (semigroup) concept, likewise the hyp-ring (hyperring) concept is the generalization of ring concept. In [2, 3], authors provided many applications of hyp-structures. There are several creators who added numerous outcomes to the hypothesis of algebraic hyp-structures, for instance, Hila and Dine [4] studied the hyperideals of left almost semi-hypergroups. Tang et al. [5] introduced the idea of hyperfilters in ordered semi-hypergroups, also see [6, 7].In 1982, Pawlak [8] introduced R-sets (rough sets) for the very first time. R-set theory has been a knowledge discovery in rational databases. Set approximation is divided into two parts, i.e., lower approximation and upper approximation. The applications of R-sets are considered in finance, pattern recognition, industries, information processing, and business. It provides a mathematical tool to find out pattern hidden in data. The major advantages of R-set approach is that it does not need any primary/secondary information about the data like the theory of probability in statistics and the grade of membership in the theory of fuzzy set. It gives systematic procedures, tools, and algorithms to find out hidden patterns in data, and it permits generating in mechanized way the sets of decision rules from data. Thivagar and Devi [9] introduced the concept of nanotopology via ring structure. R-set theory has been studied by several authors in algebraic structures and also in algebraic hyperstructures. Ahn and Kim applied R-set theory to BE-algebras [10]. Ali et al. [11] studied generalized roughness in ε,ε∨qk-fuzzy filters of ordered semigroups. Biswas and S. Nanda [12] applied R-set theory to groups. Shabir and Irshad [13] applied roughness in ordered semigroups. In [14–22], authors studied roughness in different hyperstructures. Fuzzy sets were also considered by many authors, for instance, Fotea and Davvaz [23] studied fuzzy hyperrings. Ameri and Motameni [24] applied fuzzy set theory to the hyperideals of fuzzy hyperrings. Bayrak and Yamak [25] introduced some results on the lattice of fuzzy hyperideals of a hyperring. Davvaz [26] studied fuzzy Krasner (m, n)-hyperrings. Connections between fuzzy sets and topology are considered in [27–29].


## 2. Preliminaries and Notations

Definition 1.
A topological space refers to a pairϜ,τ, where Ϝ is a nonempty set and τ is a topology on Ϝ.Definition 2.
A hyp-groupoid (hypergroupoid)ℱ,°^ is called a sm-hyp-group if, for all a,b,c of ℱ, we have a°^b°^c=a°^b°^c, which means that(1)∪dεa°^bd°^c=∪eεb°^ca°^e.Definition 3.
A subsetI of a sm-hyp-group ℱ is called right hyp-ideal (resp., left hyp-ideal) if(i)
I°^I⊆I(ii)
I°^ℱ⊆I (resp., ℱ°^I⊆I)
A left and right hyp-idealI of ℱ is known as hyp-ideal of ℱ.Definition 4.
(lower approximation of a subset, see [8]). The l-approximation (lower approximation) of Υ⊆U w.r.t E (E is an equivalence relation) is a set of all those objects, which are contained in Υ. From the diverse representations of an E-relation, we attain three productive definitions of l-approximation:(i)
ELower¯Υ=aεU:aE⊆Υ(ii)
ELower¯Υ=∪aE⊆ΥaE(iii)
ELower¯Υ=∪AεU|E:A⊆Υ, where aE=q:qEa
(i) is element-based definition, (ii) is granule-based definition, and (iii) is subsystem-based definition.Definition 5.
(upper approximation of a subset, see [8]). The u-approximation (upper approximation) of a set Υ w.r.t E is a set of all those objects which have nonempty intersection with Υ. From the unlike representations of an E-relation, we obtain three constructive definitions of u-approximation:(i)
EUpper¯Υ=aεU:aE∩Υ≠∅(ii)
EUpper¯Υ=∪aE∩Υ≠∅aE(iii)
EUpper¯Υ=∩AεU/E:A∩Υ≠∅, where aE=q:qEa
The following properties hold in approximation space [8]:(1)
ELower¯Υ⊆Υ⊆EUpper¯Υ(2)
ELower¯∅=∅=EUpper¯∅; ELower¯U=U=EUpper¯U(3)
EUpper¯Υ1∪Υ2=EUpper¯Υ1∪EUpper¯Υ2(4)
ELower¯Υ1∪Υ2⊇ELower¯Υ1∪ELower¯Υ2(5)
EUpper¯Υ1∩Υ2⊆EUpper¯Υ1∩EUpper¯Υ2(6)
ELower¯Υ1∩Υ2=ELower¯Υ1∩ELower¯Υ2(7)
Υ1⊆Υ2 implies ELower¯Υ1⊆ELower¯Υ2, EUpper¯Υ1⊆EUpper¯Υ2(8)
ELower¯¬Υ=¬EUpper¯Υ(9)
EUpper¯¬Υ=¬ELower¯Υ(10)
ELower¯ELower¯Υ=EUpper¯ELower¯Υ=ELower¯Υ(11)
EUpper¯EUpper¯Υ=ELower¯EUpper¯Υ=EUpper¯Υ


## 3. T-Structures of R-Sets Based on Sm-Hyp-Groups

In this section, we develop some concepts related to topology of R-sets based on sm-hyp-groups.Definition 6.
Letℱ be a sm-hyp-group, Υ⊆ℱ, and ξ be a REG-relation (regular relation) on ℱ. Then, the (l−) u-approximations and boundary of Υ with respect to the REG-relation ξ are given as follows:(i)
ξLower¯Υ=xεℱ:ξx⊆Υ(ii)
ξUpper¯Υ=xεℱ:ξx∩Υ≠∅(iii)
ξBΥ=ξUpper¯Υ−ξLower¯Υ
The family of sets(2)ξτΥ=ℱ,∅,ξLower¯Υ,ξUpper¯Υ,ξBΥforms a topology on ℱ.Example 1.
Letℱ=aℱ,bℱ,cℱ,dℱ be a sm-hyp-group under the binary hyperoperation “°^” defined in Cayley (Table 1).
Let(3)ξ=aℱ,aℱ,aℱ,bℱ,aℱ,cℱ,bℱ,aℱ,bℱ,bℱ,bℱ,cℱ,cℱ,aℱ,cℱ,bℱ,cℱ,cℱ,dℱ,dℱbe a REG-relation on the sm-hyp-group ℱ with the following regular classes:(4)ξaℱ=ξbℱ=ξcℱ=aℱ,bℱ,cℱandξdℱ=dℱ.
Now, letΥ=aℱ,bℱ,dℱ⊆ℱ. Then, ξLower¯Υ=dℱ, ξUpper¯Υ=ℱ, and ξBΥ=aℱ,bℱ,cℱ. Hence, ξτΥ=ℱ,∅,dℱ,aℱ,bℱ,cℱ, which is clearly a topology on ℱ.Table 1
Tabular form of the hyperoperation “°^” defined in Example 1.
°^aℱbℱcℱdℱaℱaℱbℱaℱ,cℱdℱbℱbℱbℱbℱdℱcℱaℱ,cℱbℱcℱdℱdℱdℱdℱdℱdℱRemark 1.
Letℱ be a sm-hyp-group, ξ be a REG-relation on ℱ, and Υ⊆ℱ.(i)
IfξLower¯Υ=∅ and ξUpper¯Υ=ℱ, then ξτΥ=ℱ,∅ is called the indiscrete topology on ℱ.(ii)
IfξLower¯Υ=ξUpper¯Υ=Υ, then the topology(5)ξτΥ=ℱ,∅,ξLower¯Υ=ℱ,∅,ξUpper¯Υ=ℱ,∅,Υ.(iii)
IfξLower¯Υ=∅ and ξUpper¯Υ≠ℱ, then ξτΥ=ℱ,∅,ξUpper¯Υ.(iv)
IfξLower¯Υ≠∅ and ξUpper¯Υ=ℱ, then ξτΥ=ℱ,∅,ξBΥ.(v)
IfξLower¯Υ≠ξUpper¯Υ, where ξLower¯Υ≠∅, then ξτΥ=ℱ,∅,ξLower¯Υ,ξUpper¯Υ,ξBΥ is the discrete topology on ℱ.Theorem 1.
Letℱ be a sm-hyp-group, ξ be a REG-relation on ℱ, and Υ⊆ℱ. Then,(i)
ξLower¯Υ⊆Υ⊆ξUpper¯Υ(ii)
ξLower¯∅=∅=ξUpper¯∅(iii)
ξLower¯ℱ=ℱ=ξUpper¯ℱProof.
(i)
We have to prove thatξLower¯Υ⊆Υ⊆ξUpper¯Υ. First, we prove that ξLower¯Υ⊆Υ. Let(6)xεξLower¯Υ⇒ξx⊆Υ.Asξx is a regular class of x, so xεξx. However, as ξx⊆Υ, thus xεΥ. Now, we prove that Υ⊆ξUpper¯Υ. Let yεΥ. As ξy is a regular class of y, so yεξy. Thus,(7)yεξy∩Υ⇒ξy∩Υ≠∅.Thus,yεξUpper¯Υ.(ii)
The proof of this part is straightforward.(iii)
The proof of this part is straightforward.
It is easy to see from Example1 that ξUpper¯Υ⊈Υ⊈ξLower¯Υ.Proposition 1.
Letℱ be a sm-hyp-group, ξ be a REG-relation on ℱ, and Υ1andΥ2 two subsets of ℱ such that Υ1⊆Υ2. Then,(i)
ξLower¯Υ1⊆ξLower¯Υ2(ii)
ξUpper¯Υ1⊆ξUpper¯Υ2(iii)
ξBΥ1⊆ξBΥ2Proof.
(i)
GivenΥ1⊆Υ2 and xεξLower¯Υ1, by definition(8)⇒ξx⊆Υ1for allxεℱ⇒ξx⊆Υ1⊆Υ2⇒ξx⊆Υ2for allxεℱ.Thus,ξLower¯Υ1⊆ξLower¯Υ2.(ii)
LetxεξUpper¯Υ1⇒ξx∩Υ1≠∅. Let(9)yεξx∩Υ1⇒yεξxandyεΥ1⇒yεξxandyεΥ1⊆Υ2⇒yεξx∩Υ2yεξxandyεΥ2⇒ξx∩Υ2≠∅⇒xεξUpper¯Υ2.Hence, we getξUpper¯Υ1⊆ξUpper¯Υ2.(iii)
From (i) and (ii),(10)ξUpper¯Υ1−ξLower¯Υ1⊆ξUpper¯Υ2−ξLower¯Υ2.
Thus, we haveξBΥ1⊆ξBΥ2.Theorem 2.
Letℱ be a sm-hyp-group and ξ be a REG-relation on ℱ, Υ1,Υ2⊆ℱ such that Υ1⊆Υ2. Then, ξτΥ1⊆ξτΥ2.Proof.
SinceΥ1⊆Υ2⊆ℱ, the approximations with respect to the sm-hyp-group satisfy(11)ξLower¯Υ1⊆ξLower¯Υ2,ξUpper¯Υ1⊆ξUpper¯Υ2andξBΥ1⊆ξBΥ2,which implies that ξτΥ1⊆ξτΥ2.Proposition 2.
Supposeξandγ are two REG-relations on ℱ such that ξ⊆γ, and let Υ1 be the nonempty subset of ℱ. Then,(i)
γLower¯Υ1⊆ξLower¯Υ1(ii)
ξUpper¯Υ1⊆γUpper¯Υ1(iii)
ξBΥ1⊆γBΥ1Proof.
Supposeξandγ are two REG-relations on ℱ such that ξ⊆γ, and let Υ1 be the nonempty subset of ℱ.(i)
LetxεγLower¯Υ1. Then, γx⊆Υ1. Now, as ξ⊆γ, so ξx⊆γx for any xϵℱ. Then, we get ξx⊆Υ1. Hence, xεξLower¯Υ1.(ii)
LetxεξUpper¯Υ1. Then, ξx∩Υ1≠∅. Now, as ξ⊆γ, so(12)ξx⊆γxfor anyxεℱ⇒ξx∩Υ1⊆γx∩Υ1for anyxεℱ.As∅≠ξx∩Υ1⊆γx∩Υ1. Thus, γx∩Υ1≠∅. Hence, xεγUpper¯Υ1.(iii)
The proof of this part implies from (i) and (ii).Theorem 3.
Letℱ be a sm-hyp-group and ξandγ be the REG-relations on ℱ such that ξ⊆γ, and let Υ1 be the nonempty subset of ℱ. Then, ξτΥ1≠γτΥ1.Proof.
Sinceξandγ are the REG-relations on ℱ such that ξ⊆γ, then(13)γLower¯Υ1⊆ξLower¯Υ1,ξUpper¯Υ1⊆γUpper¯Υ1andξBΥ1⊆γBΥ1,which implies that ξτΥ1≠γτΥ1.


## 4. T-Structures of R-Sets Based on Hyp-Rings

In this section, we develop some concepts related to topology of R-sets based on hyp-rings.Definition 7.
Letℜ be a hyp-ring, Υ⊆ℜ, and ℱ be a hyperideal of ℜ. Then, the (l-) u-approximations and boundary of Υ with respect to the hyp-ideal ℱ are given as follows:(i)
ℱLower¯Υ=xεℜ:x⊕ℱ⊆Υ(ii)
ℱUpper¯Υ=xεℜ:x⊕ℱ∩Υ≠∅(iii)
ℱBΥ=ℱUpper¯Υ−ℱLower¯Υ
The family of sets(14)ℱτΥ=ℜ,∅,ℱLower¯Υ,ℱUpper¯Υ,ℱBΥ,forms a topology on ℜ with respect to ℱ.Example 2.
Letℜ=aℜ,bℜ,cℜ,dℜ,eℜ,fℜ be a hyp-ring under the binary hyperoperations ⊕ and °^ defined in the Cayley (Tables 2 and 3).
Letℱ=aℜ,eℜ be a hyp-ideal of ℜ. Consider Υ=aℜ,cℜ,dℜ,fℜ⊆ℜ. Then,(15)ℱLower¯Υ=cℜ,dℜ,ℱUpper¯Υ=ℜ,ℱBΥ=aℜ,bℜ,eℜ,fℜ.
Hence,ℱτΥ=ℜ,∅,cℜ,dℜ,aℜ,bℜ,eℜ,fℜ, which is clearly a topology on ℜ.Table 2
Tabular form of the hyperoperation “⊕” defined in Example 2.
⊕aℜbℜcℜdℜeℜfℜaℜaℜbℜcℜdℜeℜfℜbℜbℜaℜ,bℜdℜcℜ,dℜfℜeℜ,fℜcℜcℜdℜaℜ,cℜ,eℜbℜ,dℜ,fℜcℜdℜdℜdℜcℜ,dℜbℜ,dℜ,fℜℜdℜcℜ,dℜeℜeℜfℜcℜdℜaℜ,eℜbℜ,fℜfℜfℜeℜ,fℜdℜcℜ,dℜbℜ,fℜaℜ,bℜ,eℜ,fℜTable 3
Tabular form of the hyperoperation “°^” defined in Example 2.
°^aℜbℜcℜdℜeℜfℜaℜaℜaℜaℜaℜaℜaℜbℜaℜbℜaℜbℜaℜbℜcℜaℜaℜcℜcℜeℜeℜdℜaℜbℜcℜdℜeℜfℜeℜaℜaℜeℜeℜaℜaℜfℜaℜbℜeℜfℜaℜbℜRemark 2.
Letℜ be a hyp-ring, ℱ be a hyp-ideal of ℜ, and Υ⊆ℜ.(i)
IfℱLower¯Υ=∅ and ℱUpper¯Υ=ℜ, then ℱτΥ=ℜ,∅ is called the indiscrete topology on ℜ.(ii)
IfℱLower¯Υ=ℱUpper¯Υ=Υ, then the topology(16)ℱτΥ=ℜ,∅,ℱLower¯Υ=ℜ,∅,ℱUpper¯Υ=ℜ,∅,Υ.(iii)
IfℱLower¯Υ=∅ and ℱUpper¯Υ≠ℜ, then ℱτΥ=ℜ,∅,ℱUpper¯Υ.(iv)
IfℱLower¯Υ≠∅ and ℱUpper¯Υ=ℜ, then ℱτΥ=ℜ,∅,ℱBΥ.(v)
IfℱLower¯Υ≠ℱUpper¯Υ where ℱLower¯Υ≠∅, then ℱτΥ=ℜ,∅,ℱLower¯Υ,ℱUpper¯Υ,ℱBΥ is the discrete topology on ℜ.Theorem 4.
Letℜ be a hyp-ring, ℱ be a hyp-ideal of ℜ, and Υ⊆ℜ. Then,(i)
ℱLower¯Υ⊆Υ⊆ℱUpper¯Υ(ii)
ℱLower¯∅=∅=ℱUpper¯∅(iii)
ℱLower¯ℜ=ℜ=ℱUpper¯ℜProposition 3.
Letℜ be a hyp-ring, ℱ be a hyp-ideal of ℜ, and Υ1andΥ2 two subsets of ℜ such that Υ1⊆Υ2. Then,(i)
ℱLower¯Υ1⊆ℱLower¯Υ2(ii)
ℱUpper¯Υ1⊆ℱUpper¯Υ2(iii)
ℱBΥ1⊆ℱBΥ2Theorem 5.
Letℜ be a hyp-ring and ℱ be a hyp-ideal of ℜ, and Υ1,Υ2⊆ℜ such that Υ1⊆Υ2. Then, ℱτΥ1⊆ℱτΥ2.Proposition 4.
Supposeℱ,W are two hyp-ideals of ℜ such that ℱ⊆W, and let Υ1 be the nonempty subset of ℜ. Then,(i)
WLower¯Υ1⊆ℱLower¯Υ1(ii)
ℱUpper¯Υ1⊆WUpper¯Υ1(iii)
ℱBΥ1⊆WBΥ1Theorem 6.
Letℜ be a hyp-ring and ℱ,W be the hyp-ideals of ℜ such that ℱ⊆W and let Υ1 be the non-empty subset of ℜ. Then ℱτΥ1≠WτΥ1.
The following theorem can also be seen in [17].Theorem 7.
LetℱandΥ2 be two hyp-ideals of ℜ. Then,(i)
ℱLower¯Υ2 is, if it is nonempty, a hyp-ideal of ℜ(ii)
ℱUpper¯Υ2 is a hyp-ideal of ℜProof.
(i)
Supposex,yεℱLower¯Υ2 and rεℜ; then,(17)x⊕ℱ⊆Υ2andy⊕ℱ⊆Υ2.This implies thatx⊕y⊕ℱ⊆Υ2 and −y⊕ℱ⊆Υ2. Also, r°^x⊕ℱ⊆Υ2 and x°^r⊕ℱ⊆Υ2. This implies that(18)x⊕y⊆ℱLower¯Υ2and−yεℱLower¯Υ2.Also,(19)r°^x⊆ℱLower¯Υ2andx°^r⊆ℱLower¯Υ2.Therefore,ℱLower¯Υ2 is a hyp-ideal of ℜ.(ii)
Supposex,yεℱUpper¯Υ2 and rεℜ; then,(20)x⊕ℱ∩Υ2≠∅andy⊕ℱ∩Υ2≠∅.
So, there exists(21)pεx⊕ℱ∩Υ2andqεy⊕ℱ∩Υ2.
SinceΥ2 is a hyp-ideal of ℜ, we have p⊕q⊆Υ2 and −qεΥ2; also,(22)p⊕q⊆x⊕ℱ⊕y⊕ℱ=x⊕y⊕ℱand−qε−y⊕ℱ.
Hence,x⊕y⊕ℱ∩Υ2≠∅ and −y⊕ℱ∩Υ2≠∅, which implies that(23)x⊕y⊆ℱUpper¯Υ2and−yεℱUpper¯Υ2.
Also, we haver⋅pεΥ2 and(24)r°^p⊆r°^x⊕ℱ=r°^x⊕ℱ.
So,r°^x⊕ℱ∩Υ2≠∅, which implies r°^x⊆ℱUpper¯Υ2. Similarly, we can prove that x°^r⊆ℱUpper¯Υ2. Therefore, ℱUpper¯Υ2 is a hyp-ideal of ℜ.Theorem 8.
LetℱandΥ2 be two hyp-ideals of ℜ. Then,(i)
ℱBΥ2 is not a hyp-ideal of ℜ if ℱLower¯Υ2≠∅(ii)
ℱBΥ2 is a hyp-ideal of ℜ if ℱLower¯Υ2=∅Corollary 1.
Letℱ,Υ be two hyp-ideals of ℜ.Then,(i)
ΥLower¯ℱ is also a hyp-ideal of ℜ, where ΥLower¯ℱ≠∅(ii)
ΥUpper¯ℱ is also a hyp-ideal of ℜ(iii)
ΥBℱ is a hyp-ideal of ℜ, when ΥLower¯ℱ=∅Theorem 9.
Letℜ and S be two hyp-rings and f be a homomorphism from ℜ to S. If Υ1 is a nonempty subset of ℜ, then(i)
fkerfUpper¯Υ1=fΥ1(ii)
fkerfLower¯Υ1⊂fΥ1Proof.
(i)
SinceΥ1⊆kerfUpper¯Υ1, it follows that fΥ1⊆fkerfUpper¯Υ1. Conversely, let yεfkerfUpper¯Υ1. Then, there exist an element xϵkerfUpper¯Υ1 such that fx=y, so we have x⊕kerf∩Υ1≠∅. Then, there exists an element aεx⊕kerf∩Υ1. Then, a=x⊕b for some bεkerf, that is, x=a−b. Then, we have(25)y=fx=fa−b=fa−fb=faεfΥ1,and sofkerfUpper¯Υ1=fΥ1.(ii)
The proof is easy.


## 5. Conclusion and Future Work

Relations between R-sets, hyp-rings, and topological structures are considered in this paper. In place of universal set, we added sm-hyp-groups and hyp-rings. In future, this work can be extended to soft set theory [30], bipolar fuzzy sets [31], intuitionistic fuzzy sets [32], or neutrosophic sets [33].



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