# Various Topologies Generated fromEj-Neighbourhoods via Ideals

Authors: Rodyna A. Hosny; Baravan A. Asaad; A. A. Azzam; Tareq M. Al-Shami
Journal: Complexity
 (2021)

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## Abstract

One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article to generate some kinds of topologies from ideals andEj-neighborhoods which are induced from any binary relation. We define new types of approximations and accuracy measures from these topologies and then compare them with their counterparts induced directly from Ej-neighborhoods and ideals. Also, we show that the approximations and accuracy measures given, herein, are better than those introduced in some previous studies under any arbitrary relation.


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## Body

## 1. Introduction

Rough set theory [1, 2]is one of the followed methods to handle the vagueness (uncertainty) of the information systems data and imperfect knowledge. In this theory, each subset is associated with two crisp sets (called lower and upper approximations) generated from an equivalence relation. To extend the applications of rough set theory, many authors have replaced an equivalence relation by different kinds of relations.The interaction between topological and rough set theory is due to Skowron [3] and Wiweger [4] who first discussed the role of topological aspects in rough sets. Then, a combination of rough set theory and topological theory became the main goal of many studies [5–11]. This interaction also included some generalizations of topology such as minimal structure [12]. We refer the reader to [13] to see the main contributions which investigated the relationships between topology and rough set theory.Ideals in a topological space have been taken into account by Kuratowski [14] and defined as a nonempty collection ℐ of subsets of a universe which is closed under finite union and subsets. Kandil et al. [15] applied the notion of ideals with r-neighborhoods to generalize Pawlak’s approximations. They showed that their results decrease the boundary region in comparison with Pawlak’s method [2], Allam’s method [16], and Yao’s method [17].Recently, some new types of neighborhood systems have been introduced and studied. Among them,Ej-neighborhood systems [18] and Cj-neighborhood systems [19] are studied. In [18], Al-Shami et al. exploited Ej-neighborhoods to establish new rough and topological approximations. They compared between them and showed that the accuracy measure obtained from rough approximations is better than their counterparts obtained from topological approximations.Through this study, we first construct new topological spaces using the ideas ofEj-neighborhood and ideals to minimize the boundary region and maximize the accuracy measure of a set compared with the approaches introduced in [18]. Second, we establish new rough approximations that improved the topological approximations.


## 2. Preliminaries

In this section, we recollect several substantial features and outcomes of rough set theory, especially those regarding to some sorts of neighborhood systems.Definition 1.
(1)
[1] A binary relation ℛ on U (i.e., ℛ⊆U×U) is said to be(i)
Equivalence if it is reflexive (i.e.,z,z∈ℛ for each z∈U), symmetric (i.e., y,z∈ℛ if z,y∈ℛ), and transitive (i.e., y,w∈ℛ, whenever y,z∈ℛ and z,w∈ℛ)(ii)
Tolerance if it is reflexive and symmetric(iii)
Preorder (quasiorder or dominance) if it is reflexive and transitive(iv)
Partial order if it is a antisymmetric (i.e.,y=z, whenever y,z∈ℛ and z,y∈ℛ) preorder(v)
Diagonal ifℛ=z,z:z∈U(vi)
Serial if everyz∈U, there exists w∈U, such that zℛw(2)
[2] For an equivalence relation ℛ on U and a subset M⊆U, the two related sets ℛ¯M=∪G∈U/ℛ:G⊆M and ℛ¯M=∪G∈U/ℛ:M∩G≠∅ are called lower approximation and upper approximation of M, respectively.Definition 2 (see [16, 17, 20]).
Letℛ be a binary relation on U. The j-neighborhoods of y∈U (Njy, in short) are defined for each j∈r,l,i,u,r,l,i,u as follows:(1)
r-neighborhood: Nry=z∈U:yℛz(2)
l-neighborhood: Nly=z∈U:yℛz(3)
i-neighborhood: Niy=Nry∩Nly(4)
u-neighborhood: Nuy=Nry∪Nly(5)
r-neighborhood: Nry=∩Nrz:y∈Nrz provided that there exists Nrz containing y. Otherwise, Nry=∅.(6)
l-neighborhood: Nly=∩Nlz:y∈Nlz provided that there exists Nlz containing y. Otherwise, Nly=∅.(7)
i-neighborhood: Niy=Nry∩Nly(8)
u-neighborhood: Nuy=Nry∪Nly
Henceforth, for all the following results, we will deal with all the values ofj, j∈r,l,i,u,r,l,i,u, unless otherwise noted.Definition 3 (see [20]).
Letℛ be a binary relation on U and ψj: U⟶PU be a mapping which assigns for each z in U its j-neighborhood in PU. Then, U,R,ψj is called a j-neighborhood space (briefly, NjS).Proposition 1 (see [18]).
LetU,ℛ,ψj be NjS and y∈U. Then,(1)
IfR is a reflexive relation, then Njy⊆Njy, for all j∈r,l,i,u(2)
IfR is a transitive relation, then Njy⊆Njy, for all j∈r,l,i,u(3)
IfR is a symmetric relation, then(1)Nry=Nly=Niy=Nuy,Nry=Nly=Niy=Nuy.Theorem 1 (see [20]).
LetU,ℛ,ψj be NjS. Then, for each j, the collection ⊤j=M⊆U:∀y∈M,Njy⊆M is a topology on U.Definition 4 (see [18]).
LetU,ℛ,ψj be NjS. Then, a set M⊆U is called j-open set if M∈⊤j, and its complement is called j-closed set. The family Γj of all j-closed sets of a j-neighborhood space is defined by Γj=F⊆U:Fc∈⊤j, where Fc is the complement of F.Definition 5 (see [18]).
Letℛ be an arbitrary binary relation on U. The Ej-neighborhood of y∈U (Ejy, in short) is defined for each j as follows:(1)
Ery=z∈U:Nrz∩Nry≠∅(2)
Ely=z∈U:Nlz∩Nly≠∅(3)
Eiy=Ery∩Ely(4)
Euy=Ery∪Ely(5)
Ery=z∈U:Nrz∩Nry≠∅(6)
Ely=z∈U:Nlz∩Nly≠∅(7)
Eiy=Ery∩Ely(8)
Euy=Ery∪ElyTheorem 2 (see [18]).
Letℛ be an arbitrary binary relation on U and y∈U. Then, Ej-neighborhoods have the following properties:(1)
y∈Ejz iff z∈Ejy(2)
Ifℛ is reflexive, then Ejy⊆Ejy and Njy⊆Ejy, for all j(3)
Ifℛ is symmetric, then Ery=Ely=Eiy=Euy and Ery=Ely=Eiy=Euy(4)
Ifℛ is transitive, then Ejy⊆Ejy, for all j∈r,l,i,u(5)
Ifℛ is symmetric and transitive, then Ejy=Njy and Ejy⊆Ejz (if y∈Ejz), for each j(6)
Ifℛ is preorder, then Ejy=Ejy, for all j∈r,l,i,u(7)
Ifℛ is an equivalence relation, then for each j, all Ejy are identical, Ejy=Njy, and Ejy=Ejz iff y∈EjzIn [18], Al-Shami et al. formulated the concepts of Ej-lower and Ej-upper approximations and Ej-accuracy measure of a subset M in terms of Ej-neighborhoods as follows:Definition 6 (see [18]).
LetM be a subset of an NjSU,ℛ,ψj. Then, the Ej-lower and Ej-upper approximations and Ej-accuracy measure of a subset M are(1)
ℛj−M=z∈U:Ejz⊆M(2)
ℛj+M=z∈U:Ejz∩M≠∅(3)
μjM=ℛj−M∩M/ℛj+M∪M, where M≠0Theorem 3 (see [18]).
LetU,ℛ,ψj be NjS and M,M´⊆U. Then, for each j, the following properties hold:(i)
∅⊆ℛj−∅ and ℛj+U⊆U(ii)
ℛj−U=U and ℛj+∅=∅(iii)
IfM⊆M´, then ℛj−M⊆ℛj−M´ and ℛj+M⊆ℛj+M´(iv)
ℛj−M∩M´=ℛj−M∩ℛj−M´ and ℛj+M∪M´=ℛj+M∪ℛj+M´(v)
ℛj−Mc=ℛj+Mc and ℛj+Mc=ℛj−Mc(vi)
ℛj−M∪ℛj−M´⊆ℛj−M∪M´ and ℛj+M∩M´⊆ℛj+M∩ℛj+M´(vii)
Generally,ℛj−ℛj−M≠ℛj−M and ℛj+ℛj+M≠ℛj+MAlso, Al-Shami et al. [18] employed Ej-neighborhoods to generate various topologies and studied their basic characteristics. Furthermore, they introduced the concepts of Ej-lower and Ej-upper approximations and Ej-accuracy measure of a subset M induced from these topologies.Theorem 4 (see [18]).
LetU,ℛ,ψj be NjS. For each j, the collection ⊤Ej=M⊆U:∀y∈M,Ejy⊆M is a topology on U.Definition 7 (see [18]).
LetU,ℛ,ψj be NjS. A set M⊆U is called Ej-open set if M∈⊤Ej, and its complement is called Ej-closed set. The family ⊥Ej of all Ej-closed sets of a j-neighborhood space is defined by ⊥Ej=F⊆U:Fc∈⊤Ej.Theorem 5 (see [18]).
Letℛ be an arbitrary binary relation on U and y∈U. If ℛ is an equivalence relation, then ⊤Er=⊤El=⊤Ei=⊤Eu=⊤Er=⊤El=⊤Ei=⊤Eu.Definition 8 (see [18]).
Let⊤Ej be a topology generated by Ej-neighborhoods. If M⊆U and j∈r,l,i,u,r,l,i,u, then the Ej-lower and Ej-upper approximations and Ej-accuracy of M are defined, respectively, as(1)
Lj⊖M=∪O∈⊤Ej:O⊆M=intEjM, where intEjM represents the interior points of M w.r.t. ⊤Ej(2)
Uj⊕M=∩F∈ℛEj:M⊆F=clEjM, where clEjM represents closure points of M w.r.t. ⊤Ej(3)
μjM=Lj⊖M/Uj⊕M, where Uj⊕M≠0Theorem 6 (see [21]).
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. If j∈r,l,i,u,r,l,i,u, then the collection ⊤jℐ=M⊆U:∀y∈M,Njy−M∈ℐ is a topology on U.


## 3. Sorts of Approximations Based onEj-Neighbourhoods and Ideals

Al-Shami et al. [18] constructed approximations relying on various topologies that are induced from the four types of Ej-neighbourhoods. In this portion, we shall generalize these topologies by using ideals and deduce new rough approximations based on Ej-neighbourhoods and ideals. We explain the relationships between these approximations and provide illustrative examples.


### 3.1. Various Topologies Generated fromEj-Neighbourhoods via Ideals

In this part, we employEj-neighborhoods and ideal ℐ to generate various topologies ζjℐ that are finer than the previous one generated by Ej-neighborhoods due to [18] for any relation.First, we are going to offer a method of generating some topologies by usingEj-neighborhoods and ideal ℐ.Theorem 7.
LetU,ℛ,ψj be NjS. For each j, the collection ζjℐ=M⊆U:∀y∈M,Ejy−M∈ℐ is a topology on U.Proof.
LetMα∈ζjℐ, α∈Δ, and z∈∪α∈ΔMα, then there exists α0∈Δ s.t. z∈Mα0. Hence, Ejz−Mα0∈ℐ. Since −∪α∈ΔMα⊆−Mα0, then Ejz−∪α∈ΔMα∈ℐ, i.e., ∪α∈ΔMα∈ζjℐ.
LetM1,M2∈ζjℐ and z∈M1∩M2. Then, Ejz−M1∈ℐ and Ejz−M2∈ℐ. According to properties of ℐ, Ejz−M1∪Ejz−M1∈ℐ. Hence, Ejz−M1∩M2∈ℐ. It follows that M1∩M2∈Ejℐ.
Easily,∅,U∈Ejℐ, ∀j. Consequently, ζjℐ is a topology on U.Definition 9.
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. A set M⊆U is called ζjℐ-open set if M∈ζjℐ, and the complement of ζjℐ-open set is called ζjℐ-closed set. The family Πjℐ of all ζjℐ-closed sets is defined by Πjℐ=F⊆U:Fc∈ζjℐ.Theorem 8.
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. Then,(1)
⊤Ej⊆ζjℐ(2)
Ifℛ is a reflexive relation and j∈r,l,i,u, then ζjℐ⊆ζjℐ(3)
Ifℛ is a symmetric, then ζrℐ=ζlℐ=ζiℐ=ζuℐ and ζrℐ=ζlℐ=ζiℐ=ζuℐ(4)
Ifℛ is a transitive relation and j∈r,l,i,u, then ζjℐ⊆ζjℐ(5)
Ifℛ is a preorder relation and j∈r,l,i,u, then ζjℐ=ζjℐ(6)
Ifℛ is an equivalence relation, then for each j, all ζjℐ are identical, ⊤jℐ=ζjℐ.Proof.
In view of Theorem2, then the proof is obvious.
The next proposition shows that the relation between the topologiesζjℐ and Ej-neighborhoods is reversible for each j.Proposition 2.
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. Then, the following results hold.(1)
ζuℐ⊆ζrℐ∩ζlℐ⊆ζrℐ∪ζlℐ⊆ζiℐ(2)
ζuℐ⊆ζrℐ∩ζlℐ⊆ζrℐ∪ζlℐ⊆ζiℐProof.
SinceEi⊆Ev⊆Eu and Ei⊆Ev⊆Eu, where v=r or l, then the proof is obvious.Example 1.
LetU=w1,w2,w3,w4 and ℛ=w1,w1,w1,w3,w1,w4,w2,w4,w4,w2,w4,w4 be a binary relation on U. In Tables 1 and 2 , we calculate Nj-neighborhoods and Ej-neighborhoods for each element of U.Table 1
Nj-neighborhoods of each element in U.
NjUw1w2w3w4Nrw1,w3,w4w4∅w2,w4Nlw1w4w1w1,w2,w4Niw1w4∅w2,w4Nuw1,w3,w4w4w1w1,w2,w4Nrw1,w3,w4w2,w4w1,w3,w4w4Nlw1w1,w2,w4∅w4Niw1w2,w4∅w4Nuw1,w3,w4w1,w2,w4w1,w3,w4w4Table 2
Ej-neighborhoods of each element in U.
EjUw1w2w3w4Erw1,w2,w4w1,w2,w4∅w1,w2,w4Elw1,w3,w4w2,w4w1,w3,w4UEiw1,w4w2,w4∅w1,w2,w4EuUw1,w2,w4w1,w3,w4UErUUUUElw1,w2w1,w2,w4∅w2,w4Eiw1,w2w1,w2,w4∅w2,w4EuUUUULemma 1.
Lety∈U and j∈r,l,i,u. Then, for any binary relation ℛ on U, Njy⊆Ejy.Proof.
Obvious.Lemma 2.
Letℐ,J be two ideals on NjSU,ℛ,ψj. If ℐ⊆J, then ζjℐ⊆ζjJ for each j.Proof.
Straightforward.
The following example shows that the inclusion in Proposition2 and Lemma 2 cannot be replaced by the equality relation.Example 2.
Continued from Example1.(2)⊤Ei=⊤El=⊤Ei=⊤Er=∅,U,w3,w1,w2,w4,⊤Eu=⊤Er=⊤Eu=⊤El=∅,U.
Ifℐ=∅,w1, then(3)ζrℐ=∅,U,w3,w2,w4,w1,w2,w4,w2,w3,w4,ζlℐ=∅,U,w2,w3,w4,ζiℐ=∅,U,w3,w2,w4,w1,w2,w4,w2,w3,w4,ζuℐ=∅,U,w2,w3,w4,ζrℐ=ζuℐ=∅,U,w2,w3,w4,ζlℐ=ζiℐ=∅,U,w3,w2,w4,w1,w2,w4,w2,w3,w4.
IfJ=∅,w1,w4,w1,w4, then(4)ζrJ=∅,U,w2,w3,w1,w2,w2,w3,w2,w4,w1,w2,w3,w1,w2,w4,w2,w3,w4,ζlJ=∅,U,w2,w3,w1,w3,w2,w3,w1,w2,w3,w2,w3,w4,ζiJ=∅,U,w1,w2,w3,w1,w2,w1,w3,w2,w3,w2,w4,w1,w2,w3,w1,w2,w4,w2,w3,w4,ζuJ=∅,U,w2,w3,w2,w3,w1,w2,w3,w2,w3,w4,ζrJ=ζuJ=∅,U,w2,w3,w1,w2,w3,w2,w3,w4,ζlJ=ζiJ=∅,U,w2,w3,w1,w2,w2,w3,w2,w4,w1,w2,w3,w1,w2,w4,w2,w3,w4.


### 3.2. Generalized Rough Approximations Based on Various Topologies Generated fromEj-Neighbourhoods via Ideals

Herein, we will construct some kinds of rough approximations using the topologies generated fromEj-neighborhoods and ideals and give some properties of them.Definition 10.
Letζjℐ be a topology generated by Ej-neighborhoods and ideal ℐ. Then, Ej-lower and Ej-upper approximations and Ej-accuracy of a subset M⊆U are defined, respectively, for each j as(1)
Ljℐ⊖M=intEjℐM, where intEjℐM represents interior of M w.r.t. ζjℐ(2)
Ujℐ⊕M=clEjℐM, where clEjℐM represents closure of M w.r.t. ζjℐ(3)
σjℐM=Ljℐ⊖M/Ujℐ⊕M, where Ujℐ⊕M≠0
Henceforth,U,ℛ,ℐ,ψj is NjS with ideal ℐ on U and denoted by EjℐS.
Several fundamental properties ofLjℐ⊖M and Ujℐ⊕M are listed in the next proposition.Proposition 3.
LetM,M´ be subsets of EjℐSU,ℛ,ℐ,ψj. Then, the following properties hold for each j.(1)
Ljℐ⊖M=Ujℐ⊕Mcc(2)
Ljℐ⊖U=U(3)
IfM⊆M´, then Ljℐ⊖M⊆Ljℐ⊖M´(4)
Ljℐ⊖M∩M´=Ljℐ⊖M∩Ljℐ⊖M´(5)
Ljℐ⊖M∪M´⊇Ljℐ⊖M∪Ljℐ⊖M´(6)
Ljℐ⊖∅=∅(7)
Ljℐ⊖M⊆M(8)
Ljℐ⊖M=Ljℐ⊖Ljℐ⊖M(9)
Ujℐ⊕M=Ljℐ⊖Mcc(10)
Ujℐ⊕∅=∅(11)
IfM⊆M´, then Ujℐ⊕M⊆Ujℐ⊕M´(12)
Ujℐ⊕M∪M´=Ujℐ⊕M∪Ujℐ⊕M´(13)
Ujℐ⊕M∩M´⊆Ujℐ⊕M∩UjℐM´(14)
Ujℐ⊕U=U(15)
M⊆Ujℐ⊕M(16)
Ujℐ⊕M=Ujℐ⊕Ujℐ⊕MAccording to Example2, Table 3 explains generalized rough approximations based on various topologies generated from Ej-neighbourhoods via ideals.Table4 demonstrates that the accuracy measure for j=i is the highest from the cases j∈r,l,i,u, and the accuracy measures for j=i,l are the highest from the case j∈r,l,i,u. However, we can find another example illustrating that the accuracy measure for j=i is the highest from the case j∈r,l,i,u.In the following remark, the inclusion relation of parts 3, 5, 7, 11, 13, and 15 in Proposition3 cannot be replaced by the equality relation.Table 3
Thej approximations for each j∈r,l,i,u,r,l,i,u.
M⊆ULrJMUrJ⊕MLlJMUlJ⊕MLiJMUiJ⊕MLuJMUuJ⊕MLrJMUrJ⊕MLlJMUlJ⊕MLiJMUiJ⊕MLuJMUuJ⊕Mw1∅w1∅w1w1w1∅w1∅w1∅w1∅w1∅w1w2w2w1,w2,w4w2w2,w4w2w2,w4w2w1,w2,w4∅w1,w2,w4w2w1,w2,w4w2w1,w2,w4∅w1,w2,w4w3w3w3w3w1,w3,w4w3w3w3w1,w3,w4∅Uw3w3w3w3∅Uw4∅w4∅w4∅w4∅w4∅w4∅w4∅w4∅w4w1,w2w1,w2w1,w2,w4w2w1,w2,w4w1,w2w1,w2,w4w2w1,w2,w4∅Uw1,w2w1,w2,w4w1,w2w1,w2,w4∅Uw1,w3w3w1,xw1,w3w1,w3,w4w1,w3w1,w3w3w1,w3,w4∅Uw3w1,w3w3w1,w3∅Uw1,w4∅w1,w4∅w1,w4w1w1,w4∅w1,w4∅w1,w4∅w1,w4∅w1,w4∅w1,w4w2,w3w2,w3Uw2,w3Uw2,w3w2,w3,w4w2,w3Uw2,w3Uw2,w3Uw2,w3Uw2,w3Uw2,w4w2,w4w1,w2,w4w2w2,w4w2,w4w2,w4w2w1,w2,w4∅Uw2,w4w1,w2,w4w2,w4w1,w2,w4∅Uw3,w4w3w3,w4w3w1,w3,w4w3w3,w4w3w1,w3,w4∅Uw3w3,w4w3w3,w4∅Uw1,w2,w3w1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w4w1,w2,w4w1,w2,w4w2w1,w2,w4w1,w2,w4w1,w2,w4w2w1,w2,w4∅Uw1,w2,w4w1,w2,w4w1,w2,w4w1,w2,w4∅Uw1,w3,w4w3w1,w3,w4w1,w3w1,w3,w4w1,w3w1,w3,w4w3w1,w3,w4∅Uw3w1,w3,w4w3w1,w3,w4∅Uw2,w3,w4w2,w3,w4Uw2,w3,w4Uw2,w3,w4w2,w3,w4w2,w3,w4Uw2,w3,w4Uw2,w3,w4Uw2,w3,w4Uw2,w3,w4UUUUUUUUUUUUUUUUUUTable 4
Comparison betweenσjℐ-accuracy measures for j∈r,l,i,u,r,l,i,u.
M⊆UσrJMσlJMσiJMσuJMσrJMσlJMσiJMσuJMw100100000w21/31/21/21/301/31/30w311/311/30110w400000000w1,w22/31/32/31/302/32/30w1,w31/22/311/301/21/20w1,w4001/200000w2,w31/21/22/31/21/21/21/21/2w2,w42/31/211/302/32/30w3,w41/21/31/21/301/21/20w1,w2,w33/43/43/43/43/43/43/43/4w1,w2,w411/311/30110w1,w3,w41/32/32/31/301/31/30w2,w3,w43/43/413/43/43/43/43/4U11111111Remark 1.
Example2 and Table 3 show that(1)
IfM=w1 and M´=w2, then LrJM⊆LrJM´ and Ujℐ⊕M⊆Ujℐ⊕M´, whereas M⊈M´.(2)
IfM=w2, M´=w4, then LiJM∪LiJM´=w2 and LiJM∪M´=w2,w4. Hence, LiJM∪M´⊈LiJM∪LiJM´.(3)
IfM=w1, then M⊈LiJM.(4)
IfM=w1,w2,w3 and M´=w1,w3,w4, then Ujℐ⊕M∩Ujℐ⊕M´=w1,w3,w4 and Ujℐ⊕M∩M´=w1,w3. Hence, Ujℐ⊕M∩Ujℐ⊕M´⊈Ujℐ⊕M∩M´.(5)
IfM=w2, then Ulℐ⊕M⊈M.Definition 11.
LetU,ℛ,ℐ,ψj be EjℐS on U. A subset M of U is called(1)
Totallyℐj definable, if Ljℐ⊖M=M=Ujℐ⊕M(2)
Internallyℐj definable, if Ljℐ⊖M=M and Ujℐ⊕M≠M(3)
Externallyℐj definable, if Ljℐ⊖M≠M and Ujℐ⊕M=M(4)
ℐj rough, if Ljℐ⊖M≠M and Ujℐ⊕M≠MRemark 2.
Example2 and Table 3 show that w3 is a totally Jr-definable set, w1,w2 is an internally Jr-definable set, w1,w4 is an externally Jr-definable set, and w1,w2 is a Jl-rough set.
The following theorem presents the relationships between the current approximations given in Definition10 and the previous one which was mentioned in Definition 8.Theorem 9.
LetU,ℛ,ℐ,ψj be EjℐS and M⊆U. Then, for each j,(1)
Lj⊖M⊆Ljℐ⊖M(2)
Ujℐ⊕M⊆Uj⊕M(3)
μjM≤σlJMProof.
It follows from the fact that⊤Ej⊆ζjℐ for each j.
According to Theorem2, the topologies ⊤jℐ and ζjℐ coincide for each j when ℛ is a symmetric and transitive relation. This means that our approximations given in Definition 10 and those given in [21] are identical under a symmetric and transitive relation. On the other hand, there is a limitation in our approaches as the next result demonstrates.Theorem 10.
LetU,ℛ,ℐ,ψj be EjℐS, such that ℛ is a reflexive relation. Then, ζjℐ⊆⊤jℐ for each j.Proof.
LetM∈ζjℐ. Then, Njy−M∈ℐ. Since Ejy⊆Njy for all j under a symmetric and transitive relation (item 2 of Theorem 2) and ℐ is an ideal, then Ejy−M∈ℐ. Hence, ζjℐ⊆⊤jℐ.


## 3.1. Various Topologies Generated fromEj-Neighbourhoods via Ideals

In this part, we employEj-neighborhoods and ideal ℐ to generate various topologies ζjℐ that are finer than the previous one generated by Ej-neighborhoods due to [18] for any relation.First, we are going to offer a method of generating some topologies by usingEj-neighborhoods and ideal ℐ.Theorem 7.
LetU,ℛ,ψj be NjS. For each j, the collection ζjℐ=M⊆U:∀y∈M,Ejy−M∈ℐ is a topology on U.Proof.
LetMα∈ζjℐ, α∈Δ, and z∈∪α∈ΔMα, then there exists α0∈Δ s.t. z∈Mα0. Hence, Ejz−Mα0∈ℐ. Since −∪α∈ΔMα⊆−Mα0, then Ejz−∪α∈ΔMα∈ℐ, i.e., ∪α∈ΔMα∈ζjℐ.
LetM1,M2∈ζjℐ and z∈M1∩M2. Then, Ejz−M1∈ℐ and Ejz−M2∈ℐ. According to properties of ℐ, Ejz−M1∪Ejz−M1∈ℐ. Hence, Ejz−M1∩M2∈ℐ. It follows that M1∩M2∈Ejℐ.
Easily,∅,U∈Ejℐ, ∀j. Consequently, ζjℐ is a topology on U.Definition 9.
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. A set M⊆U is called ζjℐ-open set if M∈ζjℐ, and the complement of ζjℐ-open set is called ζjℐ-closed set. The family Πjℐ of all ζjℐ-closed sets is defined by Πjℐ=F⊆U:Fc∈ζjℐ.Theorem 8.
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. Then,(1)
⊤Ej⊆ζjℐ(2)
Ifℛ is a reflexive relation and j∈r,l,i,u, then ζjℐ⊆ζjℐ(3)
Ifℛ is a symmetric, then ζrℐ=ζlℐ=ζiℐ=ζuℐ and ζrℐ=ζlℐ=ζiℐ=ζuℐ(4)
Ifℛ is a transitive relation and j∈r,l,i,u, then ζjℐ⊆ζjℐ(5)
Ifℛ is a preorder relation and j∈r,l,i,u, then ζjℐ=ζjℐ(6)
Ifℛ is an equivalence relation, then for each j, all ζjℐ are identical, ⊤jℐ=ζjℐ.Proof.
In view of Theorem2, then the proof is obvious.
The next proposition shows that the relation between the topologiesζjℐ and Ej-neighborhoods is reversible for each j.Proposition 2.
LetU,ℛ,ψj be NjS and ℐ be an ideal on U. Then, the following results hold.(1)
ζuℐ⊆ζrℐ∩ζlℐ⊆ζrℐ∪ζlℐ⊆ζiℐ(2)
ζuℐ⊆ζrℐ∩ζlℐ⊆ζrℐ∪ζlℐ⊆ζiℐProof.
SinceEi⊆Ev⊆Eu and Ei⊆Ev⊆Eu, where v=r or l, then the proof is obvious.Example 1.
LetU=w1,w2,w3,w4 and ℛ=w1,w1,w1,w3,w1,w4,w2,w4,w4,w2,w4,w4 be a binary relation on U. In Tables 1 and 2 , we calculate Nj-neighborhoods and Ej-neighborhoods for each element of U.Table 1
Nj-neighborhoods of each element in U.
NjUw1w2w3w4Nrw1,w3,w4w4∅w2,w4Nlw1w4w1w1,w2,w4Niw1w4∅w2,w4Nuw1,w3,w4w4w1w1,w2,w4Nrw1,w3,w4w2,w4w1,w3,w4w4Nlw1w1,w2,w4∅w4Niw1w2,w4∅w4Nuw1,w3,w4w1,w2,w4w1,w3,w4w4Table 2
Ej-neighborhoods of each element in U.
EjUw1w2w3w4Erw1,w2,w4w1,w2,w4∅w1,w2,w4Elw1,w3,w4w2,w4w1,w3,w4UEiw1,w4w2,w4∅w1,w2,w4EuUw1,w2,w4w1,w3,w4UErUUUUElw1,w2w1,w2,w4∅w2,w4Eiw1,w2w1,w2,w4∅w2,w4EuUUUULemma 1.
Lety∈U and j∈r,l,i,u. Then, for any binary relation ℛ on U, Njy⊆Ejy.Proof.
Obvious.Lemma 2.
Letℐ,J be two ideals on NjSU,ℛ,ψj. If ℐ⊆J, then ζjℐ⊆ζjJ for each j.Proof.
Straightforward.
The following example shows that the inclusion in Proposition2 and Lemma 2 cannot be replaced by the equality relation.Example 2.
Continued from Example1.(2)⊤Ei=⊤El=⊤Ei=⊤Er=∅,U,w3,w1,w2,w4,⊤Eu=⊤Er=⊤Eu=⊤El=∅,U.
Ifℐ=∅,w1, then(3)ζrℐ=∅,U,w3,w2,w4,w1,w2,w4,w2,w3,w4,ζlℐ=∅,U,w2,w3,w4,ζiℐ=∅,U,w3,w2,w4,w1,w2,w4,w2,w3,w4,ζuℐ=∅,U,w2,w3,w4,ζrℐ=ζuℐ=∅,U,w2,w3,w4,ζlℐ=ζiℐ=∅,U,w3,w2,w4,w1,w2,w4,w2,w3,w4.
IfJ=∅,w1,w4,w1,w4, then(4)ζrJ=∅,U,w2,w3,w1,w2,w2,w3,w2,w4,w1,w2,w3,w1,w2,w4,w2,w3,w4,ζlJ=∅,U,w2,w3,w1,w3,w2,w3,w1,w2,w3,w2,w3,w4,ζiJ=∅,U,w1,w2,w3,w1,w2,w1,w3,w2,w3,w2,w4,w1,w2,w3,w1,w2,w4,w2,w3,w4,ζuJ=∅,U,w2,w3,w2,w3,w1,w2,w3,w2,w3,w4,ζrJ=ζuJ=∅,U,w2,w3,w1,w2,w3,w2,w3,w4,ζlJ=ζiJ=∅,U,w2,w3,w1,w2,w2,w3,w2,w4,w1,w2,w3,w1,w2,w4,w2,w3,w4.


## 3.2. Generalized Rough Approximations Based on Various Topologies Generated fromEj-Neighbourhoods via Ideals

Herein, we will construct some kinds of rough approximations using the topologies generated fromEj-neighborhoods and ideals and give some properties of them.Definition 10.
Letζjℐ be a topology generated by Ej-neighborhoods and ideal ℐ. Then, Ej-lower and Ej-upper approximations and Ej-accuracy of a subset M⊆U are defined, respectively, for each j as(1)
Ljℐ⊖M=intEjℐM, where intEjℐM represents interior of M w.r.t. ζjℐ(2)
Ujℐ⊕M=clEjℐM, where clEjℐM represents closure of M w.r.t. ζjℐ(3)
σjℐM=Ljℐ⊖M/Ujℐ⊕M, where Ujℐ⊕M≠0
Henceforth,U,ℛ,ℐ,ψj is NjS with ideal ℐ on U and denoted by EjℐS.
Several fundamental properties ofLjℐ⊖M and Ujℐ⊕M are listed in the next proposition.Proposition 3.
LetM,M´ be subsets of EjℐSU,ℛ,ℐ,ψj. Then, the following properties hold for each j.(1)
Ljℐ⊖M=Ujℐ⊕Mcc(2)
Ljℐ⊖U=U(3)
IfM⊆M´, then Ljℐ⊖M⊆Ljℐ⊖M´(4)
Ljℐ⊖M∩M´=Ljℐ⊖M∩Ljℐ⊖M´(5)
Ljℐ⊖M∪M´⊇Ljℐ⊖M∪Ljℐ⊖M´(6)
Ljℐ⊖∅=∅(7)
Ljℐ⊖M⊆M(8)
Ljℐ⊖M=Ljℐ⊖Ljℐ⊖M(9)
Ujℐ⊕M=Ljℐ⊖Mcc(10)
Ujℐ⊕∅=∅(11)
IfM⊆M´, then Ujℐ⊕M⊆Ujℐ⊕M´(12)
Ujℐ⊕M∪M´=Ujℐ⊕M∪Ujℐ⊕M´(13)
Ujℐ⊕M∩M´⊆Ujℐ⊕M∩UjℐM´(14)
Ujℐ⊕U=U(15)
M⊆Ujℐ⊕M(16)
Ujℐ⊕M=Ujℐ⊕Ujℐ⊕MAccording to Example2, Table 3 explains generalized rough approximations based on various topologies generated from Ej-neighbourhoods via ideals.Table4 demonstrates that the accuracy measure for j=i is the highest from the cases j∈r,l,i,u, and the accuracy measures for j=i,l are the highest from the case j∈r,l,i,u. However, we can find another example illustrating that the accuracy measure for j=i is the highest from the case j∈r,l,i,u.In the following remark, the inclusion relation of parts 3, 5, 7, 11, 13, and 15 in Proposition3 cannot be replaced by the equality relation.Table 3
Thej approximations for each j∈r,l,i,u,r,l,i,u.
M⊆ULrJMUrJ⊕MLlJMUlJ⊕MLiJMUiJ⊕MLuJMUuJ⊕MLrJMUrJ⊕MLlJMUlJ⊕MLiJMUiJ⊕MLuJMUuJ⊕Mw1∅w1∅w1w1w1∅w1∅w1∅w1∅w1∅w1w2w2w1,w2,w4w2w2,w4w2w2,w4w2w1,w2,w4∅w1,w2,w4w2w1,w2,w4w2w1,w2,w4∅w1,w2,w4w3w3w3w3w1,w3,w4w3w3w3w1,w3,w4∅Uw3w3w3w3∅Uw4∅w4∅w4∅w4∅w4∅w4∅w4∅w4∅w4w1,w2w1,w2w1,w2,w4w2w1,w2,w4w1,w2w1,w2,w4w2w1,w2,w4∅Uw1,w2w1,w2,w4w1,w2w1,w2,w4∅Uw1,w3w3w1,xw1,w3w1,w3,w4w1,w3w1,w3w3w1,w3,w4∅Uw3w1,w3w3w1,w3∅Uw1,w4∅w1,w4∅w1,w4w1w1,w4∅w1,w4∅w1,w4∅w1,w4∅w1,w4∅w1,w4w2,w3w2,w3Uw2,w3Uw2,w3w2,w3,w4w2,w3Uw2,w3Uw2,w3Uw2,w3Uw2,w3Uw2,w4w2,w4w1,w2,w4w2w2,w4w2,w4w2,w4w2w1,w2,w4∅Uw2,w4w1,w2,w4w2,w4w1,w2,w4∅Uw3,w4w3w3,w4w3w1,w3,w4w3w3,w4w3w1,w3,w4∅Uw3w3,w4w3w3,w4∅Uw1,w2,w3w1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w3Uw1,w2,w4w1,w2,w4w1,w2,w4w2w1,w2,w4w1,w2,w4w1,w2,w4w2w1,w2,w4∅Uw1,w2,w4w1,w2,w4w1,w2,w4w1,w2,w4∅Uw1,w3,w4w3w1,w3,w4w1,w3w1,w3,w4w1,w3w1,w3,w4w3w1,w3,w4∅Uw3w1,w3,w4w3w1,w3,w4∅Uw2,w3,w4w2,w3,w4Uw2,w3,w4Uw2,w3,w4w2,w3,w4w2,w3,w4Uw2,w3,w4Uw2,w3,w4Uw2,w3,w4Uw2,w3,w4UUUUUUUUUUUUUUUUUUTable 4
Comparison betweenσjℐ-accuracy measures for j∈r,l,i,u,r,l,i,u.
M⊆UσrJMσlJMσiJMσuJMσrJMσlJMσiJMσuJMw100100000w21/31/21/21/301/31/30w311/311/30110w400000000w1,w22/31/32/31/302/32/30w1,w31/22/311/301/21/20w1,w4001/200000w2,w31/21/22/31/21/21/21/21/2w2,w42/31/211/302/32/30w3,w41/21/31/21/301/21/20w1,w2,w33/43/43/43/43/43/43/43/4w1,w2,w411/311/30110w1,w3,w41/32/32/31/301/31/30w2,w3,w43/43/413/43/43/43/43/4U11111111Remark 1.
Example2 and Table 3 show that(1)
IfM=w1 and M´=w2, then LrJM⊆LrJM´ and Ujℐ⊕M⊆Ujℐ⊕M´, whereas M⊈M´.(2)
IfM=w2, M´=w4, then LiJM∪LiJM´=w2 and LiJM∪M´=w2,w4. Hence, LiJM∪M´⊈LiJM∪LiJM´.(3)
IfM=w1, then M⊈LiJM.(4)
IfM=w1,w2,w3 and M´=w1,w3,w4, then Ujℐ⊕M∩Ujℐ⊕M´=w1,w3,w4 and Ujℐ⊕M∩M´=w1,w3. Hence, Ujℐ⊕M∩Ujℐ⊕M´⊈Ujℐ⊕M∩M´.(5)
IfM=w2, then Ulℐ⊕M⊈M.Definition 11.
LetU,ℛ,ℐ,ψj be EjℐS on U. A subset M of U is called(1)
Totallyℐj definable, if Ljℐ⊖M=M=Ujℐ⊕M(2)
Internallyℐj definable, if Ljℐ⊖M=M and Ujℐ⊕M≠M(3)
Externallyℐj definable, if Ljℐ⊖M≠M and Ujℐ⊕M=M(4)
ℐj rough, if Ljℐ⊖M≠M and Ujℐ⊕M≠MRemark 2.
Example2 and Table 3 show that w3 is a totally Jr-definable set, w1,w2 is an internally Jr-definable set, w1,w4 is an externally Jr-definable set, and w1,w2 is a Jl-rough set.
The following theorem presents the relationships between the current approximations given in Definition10 and the previous one which was mentioned in Definition 8.Theorem 9.
LetU,ℛ,ℐ,ψj be EjℐS and M⊆U. Then, for each j,(1)
Lj⊖M⊆Ljℐ⊖M(2)
Ujℐ⊕M⊆Uj⊕M(3)
μjM≤σlJMProof.
It follows from the fact that⊤Ej⊆ζjℐ for each j.
According to Theorem2, the topologies ⊤jℐ and ζjℐ coincide for each j when ℛ is a symmetric and transitive relation. This means that our approximations given in Definition 10 and those given in [21] are identical under a symmetric and transitive relation. On the other hand, there is a limitation in our approaches as the next result demonstrates.Theorem 10.
LetU,ℛ,ℐ,ψj be EjℐS, such that ℛ is a reflexive relation. Then, ζjℐ⊆⊤jℐ for each j.Proof.
LetM∈ζjℐ. Then, Njy−M∈ℐ. Since Ejy⊆Njy for all j under a symmetric and transitive relation (item 2 of Theorem 2) and ℐ is an ideal, then Ejy−M∈ℐ. Hence, ζjℐ⊆⊤jℐ.


## 4. Classifications of Approximations and Regions in Terms ofEj-Neighborhoods and Ideals

By using ideals andEj-neighborhoods, we deduce another method to define Ejℐ-lower and Ejℐ-upper approximations, Ejℐ-boundary region, and Ejℐ-accuracy measure of a subset M. Then, these kinds of approximations are compared with those in Section 3.2. Also, we clarify the relationships among them with the aid of several examples and show their essential properties.Definition 12.
LetM be a subset of EjℐSU,ℛ,ℐ,ψj. Then, the ℐℛj−-lower and ℐℛj+-upper approximations of a subset M are defined as(1)
ℛℐj−M=z∈U:Ejz−M∈ℐ(2)
ℛℐj+M=z∈U:Ejz∩M∉ℐ(3)
νjℐM=ℛℐj−M∩M/ℛℐj+M∪M, ℛℐj+M≠0
It should be noted that whenℐ=∅ in Definition 12, then the present approximations ℛℐj−M and ℛℐj+M coincide with the previous ones ℛℐj−M and ℛj+M in Definition 6. So, the current work is considered as a generalization of Al-Shami et al. work [18]. Now, we shall study the properties of ℛℐj−M and ℛℐj+M as it is presented in the following results.Proposition 4.
LetM be a subset of EjℐSU,ℛ,ℐ,ψj. Then, the following properties hold for each j.(i)
ℛℐu−M⊆ℛℐr−M⊆ℛℐi−M and ℛℐu−M⊆ℛℐl−M⊆ℛℐi−M(ii)
ℛℐi+M⊆ℛℐr+M⊆ℛℐu+M and ℛℐi+M⊆ℛℐl+M⊆ℛℐu+M(iii)
νuℐM≤νrℐM≤νiℐM and νuℐM≤νlℐM≤νiℐM(iv)
ℛℐu−M⊆ℛℐr−M⊆ℛℐi−M and ℛℐu−M⊆ℛℐl−M⊆ℛℐi−M(v)
ℛℐi+M⊆ℛℐr+M⊆ℛℐu+M and ℛℐi+M⊆ℛℐl+M⊆ℛℐu+M(vi)
νuℐM≤νrℐM≤νiℐM and νuℐM≤νlℐM≤νiℐMProof.
Straightforward.Theorem 11.
LetMandM´ be subsets of EjℐSU,ℛ,ℐ,ψj. Then, the following properties hold for each j.(i)
ℛℐj−U=U(ii)
IfM⊆M´, then ℛℐj−M⊆ℛℐj−M´(iii)
ℛℐj−M∩M´=ℛℐj−M∩ℛℐj−M´(iv)
ℛℐj−Mc=ℛℐj+Mc(v)
IfMc∈ℐ, then ℛℐj−M=U(vi)
ℛℐj+∅=∅(vii)
IfM⊆M´, then ℛℐj+M⊆ℛℐj+M´(viii)
ℛℐj+M∪M´=ℛℐj+M∪ℛℐj+M´(ix)
ℛℐj+Mc=ℛℐ−Mc(x)
IfM∈ℐ, then ℛℐj+M=∅Proof.
We prove only (i), (ii), (iii), (iv), and (v), and the rest of proof is similar.(i)
For eachz∈U, we have Ejz−U=∅∈ℐ; then, ℛℐj−U=U.(ii)
z∈ℛℐj−M, then Ejz−M∈ℐ. Since M⊆M´, Ejz−M´∈ℐ, i.e., z∈ℛℐj−M´. So, ℛℐj−M⊆ℛℐj−M´.(iii)
It follows from (ii) thatℛℐj−M∩M´⊆ℛℐj−M∩ℛℐj−M´. Conversely, let z∈ℛℐj−M∩ℛℐj−M´. Then, z∈ℛℐj−M and z∈ℛℐj−M´. Therefore, Ejz−M∈ℐ and Ejz−M´∈ℐ. Thus, Ejz−M∩M´∈ℐ. Hence, z∈ℛℐj−M∩M´, as required.(iv)
Sincez∈ℛℐj−Mc iff Ejz−Mc∈ℐ, iff Ejz∩M∈ℐ, iff z∉ℛℐj+M, iff z∈ℛℐj+Mc. Then, the result holds.(v)
Obvious
In view of (ii) and (vii) of Theorem11, the next corollary is obvious.Corollary 1.
LetMandM´ be subsets of EjℐSU,ℛ,ℐ,ψj. Then, the following properties hold for each j.(i)
ℛℐj−M∪ℛℐj−M´⊆ℛℐj−M∪M´(ii)
ℛℐj+M∩M´⊆ℛℐj+M∩ℛℐj+M´Example 3.
In view of example1, Table 5 offers japproximations for each j.
According to Table5, we construct Table 6 which represents the νjℐ-accuracy measure for every subsets of U.
To elucidate that the converses of the items (ii), (v), (vii), and (x) of Theorem11 and Corollary 1 are not always credible, we display the next example.Table 5
Thej approximations for each j∈r,l,i,u,r,l,i,u.
M⊆UℛJr−MℛJr+MℛJl−MℛJl+MℛJi−MℛJi+MℛJu−MℛJu+MℛJr−MℛJr+MℛJl−MℛJl−MℛJi−MℛJi+MℛJu−MℛJr+Mw1w3∅∅∅w1,w3∅∅∅∅∅w3∅w3∅∅∅w2Uw1,w2,w4w2w2,w4Uw2,w4w2w1,w2,w4∅UUw1,w2,w4Uw1,w2,w4∅Uw3w3∅w1,w3w1,w3,w4w1,w3∅w3w1,w3,w4∅Uw3∅w3∅∅Uw4w3∅∅∅w1,w3∅∅∅∅∅w3∅w3∅∅∅w1,w2Uw1,w2,w4w2w2,w4Uw2,w4w2w1,w2,w4∅UUw1,w2,w4Uw1,w2,w4∅Uw1,w3w3∅w1,w3w1,w3,w4w1,w3∅w3w1,w3,w4∅Uw3∅w3∅∅Uw1,w4w3∅∅∅w1,w3∅∅∅∅∅w3∅w3∅∅∅w2,w3Uw1,w2,w4UUUw2,w4UUUUUw1,w2,w4Uw1,w2,w4UUw2,w4Uw1,w2,w4w2w2,w4Uw2,w4w2w1,w2,w4∅UUw1,w2,w4Uw1,w2,w4∅Uw3,w4w3∅w1,w3w1,w3,w4w1,w3∅w3w1,w3,w4∅Uw3∅w3∅∅Uw1,w2,w3Uw1,w2,w4UUUw2,w4UUUUUw1,w2,w4Uw1,w2,w4UUw1,w2,w4Uw1,w2,w4w2w2,w4Uw2,w4w2w1,w2,w4∅UUw1,w2,w4Uw1,w2,w4∅Uw1,w3,w4w3∅w1,w3w1,w3,w4w1,w3∅w3w1,w3,w4∅Uw3∅w3∅∅Uw2,w3,w4Uw1,w2,w4UUUw2,w4UUUUUw1,w2,w4Uw1,w2,w4UUUUw1,w2,w4UUUw2,w4UUUUUw1,w2,w4Uw1,w2,w4UU∅w3∅∅∅w1,w3∅∅∅∅∅w3∅w3∅∅∅Table 6
Comparison betweenνjℐ-accuracy measures for j∈r,l,i,u,r,l,i,u.
M⊆UνrJMνlJMνiJMνuJMνrJMνlJMνiJMνuJMw100100000w21/31/21/21/301/31/30w311/311/30110w400000000w1,w22/31/32/31/302/32/30w1,w31/22/311/301/21/20w1,w4001/200000w2,w31/21/22/31/21/21/21/21/2w2,w42/31/211/302/32/30w3,w41/21/31/21/301/21/20w1,w2,w33/43/43/43/43/43/43/43/4w1,w2,w411/311/30110w1,w3,w41/32/32/31/301/31/30w2,w3,w43/43/413/43/43/43/43/4U11111111Example 4.
In view of Example1 and Table 5, note the following.(i)
ForM=w3 and M´=w2,w4, we have ℛℐr−M=w3 and ℛℐr−M´=U. But, M⊈M´.(ii)
ForM=w2,w4, ℛℐr−M=U. But, Mc∉ℐ.(iii)
ForM=w1, we find M´=w2, we have ℛℐu+M=∅ and ℛℐu+M´=w1,w2,w4. But, M⊈M´.(iv)
ForM=w3, ℛℐr+M=∅. But, M∉ℐ.(v)
ForM=w2 and M´=w4, we have ℛℐl−M=w1,w3, ℛℐl−M´=w1,w3, and ℛℐl−S∪M´=U. Then, ℛℐl−M∪ℛℐl−M´≠ℛℐl−S∪M´.(vi)
ForM=w1,w3 and M´=w1,w4, we have ℛℐr+M=U, ℛℐr+M´=U, and ℛℐr+S∩M´=∅. Then, ℛℐr+M∩ℛℐr+M´≠ℛℐr+S∩M´.
Note that some basic properties of rough sets with respect toℛℐj−-lower and ℛℐj+-upper approximations may evaporate. In what follows, we mention those missing properties.(i)
ℛℐj−M⊆M(ii)
ℛℐj−∅=∅(iii)
ℛℐj−ℛℐj−M=ℛℐj−M(iv)
M⊆ℛℐj+M(v)
ℛℐj+U=U(vi)
ℛℐj+ℛℐj+M=ℛℐj+M
The next example supports assertions of the above note.Example 5.
In view of Example1 and Table 5, note the following.(i)
IfM=w1, then ℛℐi−M=w3. Hence, ℛℐi−M⊈M.(ii)
ℛℐr−∅=w3≠∅.(iii)
IfM=w2,w4, then ℛℐu−M=w2 and ℛℐu−ℛℐu−M=∅. Hence, ℛℐu−ℛℐu−M≠ℛℐu−M.(iv)
IfM=w1, then ℛℐl+M=∅. Hence, M⊈ℛℐl+M.(v)
ℛℐi+U=w1,w2,w4≠U.(vi)
IfM=w2, then ℛℐu+M=w1,w2,w4 and ℛℐj+ℛℐj+M=U. Hence, ℛℐj+ℛℐj+M≠ℛℐj+M.Lemma 3.
LetℐandJ be two ideals on U,ℛ,ψj and M⊆U. If ℐ⊆J, then the following statements hold for each j:(1)
ℛℐj−M⊆ℛ−jJM(2)
ℛℐj+M⊇ℛ¯jJM(3)
νjℐM≤νjJMProof.
Straightforward.
The next theorem shows the relationships between the approximationsℛℐj−M and ℛℐj+M of the current approximations in Definition 6 and ℛj−M and ℛj+M of Definition 6 for each j.Theorem 12.
LetU,ℛ,ψj,ℐ and M⊆U. Then, the following statements hold for each j:(1)
ℛj−M⊆ℛℐj−M(2)
ℛℐj+M⊆ℛj+M(3)
μjM≤νjℐMProof.
Direct to prove.
To preserve some Pawalk’s properties which are loss in the approximations given in Definition12, we formulate new approximations and accuracy measures in the following.Definition 13.
LetM be a subset of EjℐSU,ℛ,ℐ,ψj. Then, the ℛj⋄-lower and ℛj⋄-upper approximations and νjℐ-accuracy measure of a subset M are(1)
ℛj⋄M=ℛℐj−M∩M(2)
ℛj⋄M=ℛℐj+M∪M(3)
νj⋄ℐM=ℛj⋄M/ℛj⋄M, ℛj⋄M≠0.Remark 3.
It should be noted that the current approximationsℛj⋄M and ℛj⋄M in Definition 13 have the same properties of the current approximations ℛℐj−M and ℛℐj+M, which are stated in Theorem 11 and Corollary 1. Additionally, it satisfies the following properties:(1)
ℛj⋄M⊆M⊆ℛj⋄M(2)
ℛj⋄∅=∅, ℛj⋄U=U.


## 5. Conclusion and Future Work

Rough set is a powerful mathematical to deal with uncertainty. Approximation operators are the core concepts in rough set content; they have topological properties similar to all/some properties of the interior and closure operators. Neighborhood systems are one of the followed methods to study rough approximations using topological interior and closure operators.In this study, we have initiated different types of topologies from ideals andEj-neighborhoods induced from any binary relations. We have applied these topologies to study new kinds of approximations and accuracy measures. Then, we have compared between them and their counterparts induced directly from Ej-neighborhoods and ideals. Also, we have illustrated the advantages of our approaches to obtain higher accuracy measures than those proposed in [18]. Some limitations of our approaches have been investigated.In the upcoming works, we will study new types of topologies and approximations induced from other neighborhoods and ideals. Also, we will investigate the concepts and results presented, herein, in fuzzy and rough sets contents.



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