# Some Topological Approaches for Generalized Rough Sets via Ideals

Authors: Tareq M. Al-shami; Hüseyin Işık; Ashraf S. Nawar; Rodyna A. Hosny
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts ofj-adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.


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## Body

## 1. Introduction

The idea of rough set theory (RST) was first put forth by Pawlak [1], where imperfect information gives rise to indiscernibility of objects. RST proves its adequacy to treat and model a lot of real-life issues that were constructed as a method to overcome imperfectness and ambiguity of information. The classical RST was characterized by a pair of approximation operators called lower and upper approximations which are established using equivalence classes. But at times, equivalence relations are tricky to be acquired in real-life issues due to the incompleteness of human information. So, a lot of ideas and articles have been introduced to generalizing the classical theory of RSs, for further specifics, see reference [2]. For interpretation of the granules, both Lin [3] and Yao [4] studied the RSs utilizing neighborhood systems. In fact, they freed RST from an equivalence relation which is a very inflexible obligation that restricts the real-life implementation scope of rough sets philosophy.Quite recently, Abd El-Monsef et al. [5] applied j-neighborhoods to generalize the classical RST. Allam et al. presented the concepts of minimal right neighborhoods and minimal left neighborhoods in [6, 7], respectively. Dai et al. [8] presented new rough set models using maximal neighborhoods induced from similarity relations. Al-shami [9] defined containment neighborhoods and applied to protect medical staff from infected diseases. Also, Al-shami et al. [10] initiated several types of lower and upper approximations using Nj-neighborhoods. In [11], El-Bably and Fleifel investigated new topological structures by relations. El-Bably et al. [12] introduced some closure operators using arbitrary binary relation and generated some topologies from any binary relation without using sub-base or base.Study of the rough set theory via topology is an enjoyable topic that received the attention of many researchers, see, for example [2, 13–19]. An ideal ℐ [20] on a nonempty finite set (universe) U is a nonempty family of subsets of U with heredity property, as well as it is closed under finite unions. Some interesting papers studied ideals via rough set theory such as [15, 21]. Hosny [22] replaced Pawlak’s approximations (lower and upper) by topological operators (interior and closure). Then, she [23] presented different methods to establish new rough set models using ideals. In 2020, Kandil et al. [24] offered the collection ℐ,ℐ^ generated by two ideals ℐ and ℐ^ and proved that it is also an ideal on a universe.From topology’s view, our work discusses the notions of rough sets based on the topological spaces generated byj-adhesion neighborhoods and ideals. That is, it studies the concepts of lower and upper approximations in terms of j-adhesion neighborhoods via ideals. This approach minimizes the vagueness of uncertainty regions at their borders by increasing the lower approximation and decreasing the upper approximation which automatically implies increasing the accuracy measure of the uncertainty regions.The layout of this paper is as follows. In Section2, we recall three classes of neighborhood systems called Nj-neighborhoods, Ej-neighborhoods, and Pj-neighborhoods as well as the results that show the way of generating topologies using these classes of neighborhoods. In Section 3, we apply the notions of ideal and Pj-neighborhood systems to establish new types of topologies. Based on these topologies, we propose new kind of rough set models and compare them with the previous ones under reflexive and arbitrary relations. We devote Section 4 to discuss the approximations and accuracy measures generated by Ej-neighborhoods and Pj-neighborhoods with ideals. Eventually, Section 5 gives conclusions and some directions for future works.


## 2. Preliminaries

In this part, we present thej-neighborhood space, which depends on a finite number of various kinds of arbitrary binary relations. So, we produce eight diverse topologies and investigate the relationships among these topologies. Then, we get eight techniques to find the lower and upper approximations of rough sets. Comparisons between the accuracy of these types of new approximations are attained.Henceforward, we considerj∈r,l,r,l,i,u,i,u, unless otherwise specified.


### 2.1. Approximations and Topologies Generated by Differentj-Neighborhoods

Definition 1 (see [5]).
LetR be a binary relation on U. The eight sorts of j-neighborhoods of any point h∈U, say Njh, are given by(1)
Nrh = z∈U:hRz.(2)
Nlh = z∈U:zRh.(3)
Nih = Nrh∩Nlh.(4)
Nuh = Nrh∪Nlh.(5)
Nrh = ∩Nrz:h∈Nrz; if Nrz does not exist there such that h∈Nrz, then Nrh=∅.(6)
Nlh = ∩Nlz:h∈Nlz; if Nlz does not exist there such that h∈Nlz, then Nlh=∅.(7)
Nih = Nrh∩Nlh.(8)
Nuh = Nrh∪Nlh.Definition 2 (see [5]).
The tripleU,R,ψj is called a j-neighborhood space (in short, NjS), where ψj is a mapping from U to PU which assigns for each point in U its j-neighborhood.Proposition 1 (see [25]).
LetU,R,ψj be a NjS and h∈U. Then,(1)
The reflexivity ofR implies that Njh is a nonempty set.(2)
The reflexivity ofR implies that Njh is a subset of Njh, for every j∈r,l,i,u.(3)
IfR is a transitive relation, then Njh is a subset of Njh, for every j∈r,l,i,u.(4)
The symmetry ofR implies that Nrh = Nlh = Nih = Nuh and Nrh = Nlh = Nih = Nuh.Theorem 1 (see [5]).
LetU,R,ψj be a NjS. Then, the family ⊤j = {M⊆U: ∀h∈M, Njh⊆M} forms a topology on U.Definition 3 (see [5]).
LetU,R,ψj be a NjS. Then, we called a subset M⊆U a j-open set if M∈⊤j, and we called its complement a j-closed set. The class Γj of all j-closed sets of a j-neighborhood space is given by Γj = F⊆U:Fc∈⊤j.Definition 4 (see [5]).
Let⊤j be a topology induced from j-neighborhoods. The j-lower, j-upper approximations, and j-accuracy of M⊆U are respectively given by(1)
R¯jM = ∪O∈⊤j:O⊆M = intjM.(2)
R¯jM = ∩F∈Γj:M⊆F = cljM.(3)
αjM = |R¯jM|/R¯jM, where R¯jM≠0.


### 2.2. Approximations and Topologies Generated by DifferentEj-Neighborhoods

Definition 5 (see [26]).
LetR be a binary relation on U. The Ej-neighborhood of a point h∈U (briefly, Ejh) is formulated as(1)
Erh = {z∈U: the intersection of Nrz and Nrh is nonempty}.(2)
Elh = {z∈U: the intersection of Nlz and Nlh is nonempty}.(3)
Eih = Erh∩Elh.(4)
Euh = Erh∪Elh.(5)
Erh = {z∈U: the intersection of Nrz and Nrh is nonempty}.(6)
Elh = {z∈U: the intersection of Nlz and Nlh is nonempty}.(7)
Eih = Erh∩Elh.(8)
Euh = Erh∪Elh.Theorem 2 (see [26]).
LetR be a binary relation on U and h∈U. Then,(1)
h∈Ejz iff z∈Ejh.(2)
The reflexivity ofℛ implies that Ejh is a subset of Ejh and Njh is a subset of Ejh for every j.(3)
The symmetry ofℛ implies that Erh, Elh, Eih, and Euh are equal, and Erh, Elh, Eih, and Euh are equal.(4)
The transitivity ofℛ implies that Ejh is a subset of Ejh for each j∈u,i,r,l.(5)
Ifℛ is transitive and symmetric, then Ejh = Njh and Ejh⊆Ejz (if h∈Ejz), for each j.(6)
Ifℛ is preorder, then Ejh=Ejh, for all j∈r,l,i,u.(7)
Ifℛ is an equivalence, then Ejh are equal for each j, and Ejh = Njh.Theorem 3 (see [26]).
For eachj, the family ⊤Ej = {M⊆U: ∀h∈M, Ejh⊆M} forms a topology on U.Definition 6 (see [26]).
LetU,R,ψj be an EjS. We called a set M⊆U an Ej-open set if M∈⊤Ej, and we called its complement an Ej-closed set. The class ΓEj of all Ej-closed sets is given by ΓEj =  K⊆U:Kc∈⊤Ej.Definition 7 (see [26]).
Consider⊤Ej as a topology induced by Ej neighborhoods. For each j, the Ej lower, Ej upper approximations, and Ej accuracy of M⊆U are respectively given by(1)
Lj⊖M = ∪O∈⊤Ej:O⊆M = intEjM.(2)
Uj⊕M = ∩F∈ΓEj:M⊆F = clEjM.(3)
μjM = |Lj⊖M|/Uj⊕M, where Uj⊕M≠0.


### 2.3. Approximations and Topologies Generated by Differentj-Adhesion Neighborhoods

Definition 8 (see [27]).
LetR be a binary relation on U. The j-adhesion (Pj) neighborhood of any point h∈U (denoted by Pjh) is defined as(1)
r-adhesion neighborhood: Prh = {z∈U:Nrz is equal to Nrh}.(2)
l-adhesion neighborhood: Plh = {z∈U:Nlz is equal to Nlh}.(3)
i-adhesion neighborhood: Pih = Prh∩Plh.(4)
u-adhesion neighborhood: Puh = Prh∪Plh.(5)
r-adhesion neighborhood: Prh = {z∈U:Nrz is equal to Nrh}.(6)
l-adhesion neighborhood: Plh = {z∈U:Nlz is equal to Nlh}.(7)
i-adhesion neighborhood Pih = Prh∩Plh.(8)
u-adhesion neighborhood: Puh = Prh∪Plh.Remark 1.
It should be noted that the concept ofPj-adhesion neighborhood of any point in U in [27] is the same as the notion of core of neighborhood systems induced by R in [28].Proposition 2 (see [25]).
LetR be a binary relation on U and h∈U. Then, Pj-neighborhoods have the next properties:(1)
Prh = Plh, Plh = Prh.(2)
Pih = Pih, Puh = Puh.Lemma 1 (see [25]).
LetR be a binary relation on U and h,z∈U. Then,(1)
h∈Pjh for all j.(2)
z∈Pjh iff Pjz = Pjh for every j∈i,i,r,r,l,l.Corollary 1 (see [25]).
LetR be a binary relation on U. The class ℘U = Pjy|y∈U forms a partition for U in the cases of j∈i,i,r,r,l,l.Proposition 3 (see [25]).
ConsiderR as a reflexive relation on U and h∈U. Then, Pjh⊆Njh⊆Ejh for each j.Proposition 4 (see [25]).
ConsiderR as an equivalence relation on U and h∈U. Then, Pjh=Njh=Ejh for each j.Theorem 4 (see [25]).
LetU,R,ψj be a j-adhesion neighborhood space (PjS). Then, for each j∈r,l,i,u, the class ⊤Pj = {M⊆U: ∀h∈M, Pjh⊆M} forms a topology on U.Definition 9 (see [25]).
LetU,R,ψj be a j-adhesion neighborhood space. We called a set M⊆U a j-adhesion open set if M∈⊤Pj, and we called its complement a j-adhesion closed set. The family ΓPj of all j-adhesion closed sets of a j-neighborhood space is defined by ΓPj = K⊆U:Kc∈⊤Pj.Remark 2 (see [25]).
LetU,R,ψj be a j-adhesion neighborhood space. Then,(1)
The collection⊤Pj is a quasidiscrete (clopen) topology on U, ∀j∈i,i,r,r,l,l.(2)
For eachj, ⊤j⊆⊤Pj provided that R is reflexive.Definition 10 (see [25]).
LetU,R,ψj be a PjS and ⊤Pj be a topology generated by j-adhesion neighborhoods. The j-adhesion lower, j-adhesion upper approximations, and j-adhesion accuracy of M⊆U are respectively given by(1)
LjM = ∪O∈⊤Pj:O⊆M = intPjM.(2)
UjM = ∩F∈ΓPj:M⊆F = clPjM.(3)
θjM = |LjM|/UjM, where UjM≠0.


### 2.4. Approximations and Topologies Generated by Differentj-Neighborhoods and Ideals

In [23], Hosny presented an idea depending on generating different topologies by using j-neighborhoods and ideals and studied some of their properties.Theorem 5 (see [23]).
ConsiderU,R,ψj as a NjS and ℐ as an ideal on U. Then, the family ⊤jℐ = {M⊆U: ∀h∈M, Njh − M∈ℐ} forms a topology on U for each j.Theorem 6 (see [23]).
LetU,R,ψj be a NjS and ℐ be an ideal on U. Then, ⊤j⊆⊤jℐ for each j.Definition 11 (see [23]).
LetU,R,ψj be a NjS and ℐ be an ideal on U. We called a set M⊆U an ℐj-open set if M∈⊤jℐ, and we called its complement an ℐj-closed set. The family Γjℐ of all ℐj-closed sets of a j-neighborhood space is given by Γjℐ = K⊆U:Kc∈⊤jℐ.Definition 12 (see [23]).
LetU,R,ψj be a NjS and ℐ be an ideal on U. The ℐj-lower, ℐj-upper approximations, and ℐj-accuracy of the approximation of M⊆U are respectively given by(1)
R¯jℐM = ∪O∈⊤jℐ:O⊆M = intjℐM, where intjℐM is an ℐj interior of M.(2)
R¯jℐM = ∩F∈Γjℐ:M⊆F = cljℐM, where cljℐM is an ℐj closure of M.(3)
αjℐM = |R¯jℐM|/R¯jℐM, where R¯jℐM≠0.


### 2.5. Approximations and Topologies Generated by DifferentEj-Neighborhoods and Ideals

For any binary relation, Hosny et al. [29] utilized the concepts of Ej-neighborhoods and ideal ℐ to output various topologies ζjℐ which are finer than the previous one generated by Ej-neighborhoods due to [26].Theorem 7 (see [29]).
The classζjℐ = {M⊆U: ∀h∈M, Ejh−M∈ℐ} forms a topology on U for each j.Definition 13 (see [29]).
LetU,R,ψj be a EjS and ℐ be an ideal on U. We called a set M⊆U a ζjℐ-open set if M∈ζjℐ, and we called its complement a ζjℐ-closed set. The class Πjℐ of all ζjℐ-closed sets is given by Πjℐ = K⊆U:Kc∈ζjℐ.Theorem 8 (see [29]).
ConsiderU,R,ψj as an EjS and ℐ as an ideal on U. Then,(1)
⊤Ej⊆ζjℐ.(2)
The reflexivity ofℛ implies that ζjℐ⊆ζjℐ for each j∈i,u,r,l.(3)
Ifℛ is a symmetric, then ζrℐ = ζlℐ = ζiℐ = ζuℐ and ζrℐ = ζlℐ = ζiℐ = ζuℐ.(4)
The transitivity ofℛ implies that ζjℐ⊆ζjℐ for each j∈i,u,r,l.(5)
Ifℛ is a preorder, then ζjℐ=ζjℐ for each j∈i,u,r,l.(6)
Ifℛ is an equivalence, then all ζjℐ are equal, and ⊤jℐ = ζjℐ for each j.Lemma 2 (see [29]).
For any binary relationℛ on U, we have Njh⊆Ejh for each h∈U and j∈i,u,r,l.Definition 14 (see [29]).
Considerζjℐ as a topology generated by Ej-neighborhoods and ideal ℐ. Then, for each j, ℐEj-lower, ℐEj-upper approximations, and ℐEj-accuracy of a subset M⊆U are defined respectively as follows:(1)
Ljℐ⊖M =  intEjℐM, where intEjℐM forms the interior points of M in ζjℐ.(2)
Ujℐ⊕M = clEjℐM, where clEjℐM forms the closure points of M in ζjℐ.(3)
σjℐM = |Ljℐ⊖M|/Ujℐ⊕M, where Ujℐ⊕M≠0.


## 2.1. Approximations and Topologies Generated by Differentj-Neighborhoods

Definition 1 (see [5]).
LetR be a binary relation on U. The eight sorts of j-neighborhoods of any point h∈U, say Njh, are given by(1)
Nrh = z∈U:hRz.(2)
Nlh = z∈U:zRh.(3)
Nih = Nrh∩Nlh.(4)
Nuh = Nrh∪Nlh.(5)
Nrh = ∩Nrz:h∈Nrz; if Nrz does not exist there such that h∈Nrz, then Nrh=∅.(6)
Nlh = ∩Nlz:h∈Nlz; if Nlz does not exist there such that h∈Nlz, then Nlh=∅.(7)
Nih = Nrh∩Nlh.(8)
Nuh = Nrh∪Nlh.Definition 2 (see [5]).
The tripleU,R,ψj is called a j-neighborhood space (in short, NjS), where ψj is a mapping from U to PU which assigns for each point in U its j-neighborhood.Proposition 1 (see [25]).
LetU,R,ψj be a NjS and h∈U. Then,(1)
The reflexivity ofR implies that Njh is a nonempty set.(2)
The reflexivity ofR implies that Njh is a subset of Njh, for every j∈r,l,i,u.(3)
IfR is a transitive relation, then Njh is a subset of Njh, for every j∈r,l,i,u.(4)
The symmetry ofR implies that Nrh = Nlh = Nih = Nuh and Nrh = Nlh = Nih = Nuh.Theorem 1 (see [5]).
LetU,R,ψj be a NjS. Then, the family ⊤j = {M⊆U: ∀h∈M, Njh⊆M} forms a topology on U.Definition 3 (see [5]).
LetU,R,ψj be a NjS. Then, we called a subset M⊆U a j-open set if M∈⊤j, and we called its complement a j-closed set. The class Γj of all j-closed sets of a j-neighborhood space is given by Γj = F⊆U:Fc∈⊤j.Definition 4 (see [5]).
Let⊤j be a topology induced from j-neighborhoods. The j-lower, j-upper approximations, and j-accuracy of M⊆U are respectively given by(1)
R¯jM = ∪O∈⊤j:O⊆M = intjM.(2)
R¯jM = ∩F∈Γj:M⊆F = cljM.(3)
αjM = |R¯jM|/R¯jM, where R¯jM≠0.


## 2.2. Approximations and Topologies Generated by DifferentEj-Neighborhoods

Definition 5 (see [26]).
LetR be a binary relation on U. The Ej-neighborhood of a point h∈U (briefly, Ejh) is formulated as(1)
Erh = {z∈U: the intersection of Nrz and Nrh is nonempty}.(2)
Elh = {z∈U: the intersection of Nlz and Nlh is nonempty}.(3)
Eih = Erh∩Elh.(4)
Euh = Erh∪Elh.(5)
Erh = {z∈U: the intersection of Nrz and Nrh is nonempty}.(6)
Elh = {z∈U: the intersection of Nlz and Nlh is nonempty}.(7)
Eih = Erh∩Elh.(8)
Euh = Erh∪Elh.Theorem 2 (see [26]).
LetR be a binary relation on U and h∈U. Then,(1)
h∈Ejz iff z∈Ejh.(2)
The reflexivity ofℛ implies that Ejh is a subset of Ejh and Njh is a subset of Ejh for every j.(3)
The symmetry ofℛ implies that Erh, Elh, Eih, and Euh are equal, and Erh, Elh, Eih, and Euh are equal.(4)
The transitivity ofℛ implies that Ejh is a subset of Ejh for each j∈u,i,r,l.(5)
Ifℛ is transitive and symmetric, then Ejh = Njh and Ejh⊆Ejz (if h∈Ejz), for each j.(6)
Ifℛ is preorder, then Ejh=Ejh, for all j∈r,l,i,u.(7)
Ifℛ is an equivalence, then Ejh are equal for each j, and Ejh = Njh.Theorem 3 (see [26]).
For eachj, the family ⊤Ej = {M⊆U: ∀h∈M, Ejh⊆M} forms a topology on U.Definition 6 (see [26]).
LetU,R,ψj be an EjS. We called a set M⊆U an Ej-open set if M∈⊤Ej, and we called its complement an Ej-closed set. The class ΓEj of all Ej-closed sets is given by ΓEj =  K⊆U:Kc∈⊤Ej.Definition 7 (see [26]).
Consider⊤Ej as a topology induced by Ej neighborhoods. For each j, the Ej lower, Ej upper approximations, and Ej accuracy of M⊆U are respectively given by(1)
Lj⊖M = ∪O∈⊤Ej:O⊆M = intEjM.(2)
Uj⊕M = ∩F∈ΓEj:M⊆F = clEjM.(3)
μjM = |Lj⊖M|/Uj⊕M, where Uj⊕M≠0.


## 2.3. Approximations and Topologies Generated by Differentj-Adhesion Neighborhoods

Definition 8 (see [27]).
LetR be a binary relation on U. The j-adhesion (Pj) neighborhood of any point h∈U (denoted by Pjh) is defined as(1)
r-adhesion neighborhood: Prh = {z∈U:Nrz is equal to Nrh}.(2)
l-adhesion neighborhood: Plh = {z∈U:Nlz is equal to Nlh}.(3)
i-adhesion neighborhood: Pih = Prh∩Plh.(4)
u-adhesion neighborhood: Puh = Prh∪Plh.(5)
r-adhesion neighborhood: Prh = {z∈U:Nrz is equal to Nrh}.(6)
l-adhesion neighborhood: Plh = {z∈U:Nlz is equal to Nlh}.(7)
i-adhesion neighborhood Pih = Prh∩Plh.(8)
u-adhesion neighborhood: Puh = Prh∪Plh.Remark 1.
It should be noted that the concept ofPj-adhesion neighborhood of any point in U in [27] is the same as the notion of core of neighborhood systems induced by R in [28].Proposition 2 (see [25]).
LetR be a binary relation on U and h∈U. Then, Pj-neighborhoods have the next properties:(1)
Prh = Plh, Plh = Prh.(2)
Pih = Pih, Puh = Puh.Lemma 1 (see [25]).
LetR be a binary relation on U and h,z∈U. Then,(1)
h∈Pjh for all j.(2)
z∈Pjh iff Pjz = Pjh for every j∈i,i,r,r,l,l.Corollary 1 (see [25]).
LetR be a binary relation on U. The class ℘U = Pjy|y∈U forms a partition for U in the cases of j∈i,i,r,r,l,l.Proposition 3 (see [25]).
ConsiderR as a reflexive relation on U and h∈U. Then, Pjh⊆Njh⊆Ejh for each j.Proposition 4 (see [25]).
ConsiderR as an equivalence relation on U and h∈U. Then, Pjh=Njh=Ejh for each j.Theorem 4 (see [25]).
LetU,R,ψj be a j-adhesion neighborhood space (PjS). Then, for each j∈r,l,i,u, the class ⊤Pj = {M⊆U: ∀h∈M, Pjh⊆M} forms a topology on U.Definition 9 (see [25]).
LetU,R,ψj be a j-adhesion neighborhood space. We called a set M⊆U a j-adhesion open set if M∈⊤Pj, and we called its complement a j-adhesion closed set. The family ΓPj of all j-adhesion closed sets of a j-neighborhood space is defined by ΓPj = K⊆U:Kc∈⊤Pj.Remark 2 (see [25]).
LetU,R,ψj be a j-adhesion neighborhood space. Then,(1)
The collection⊤Pj is a quasidiscrete (clopen) topology on U, ∀j∈i,i,r,r,l,l.(2)
For eachj, ⊤j⊆⊤Pj provided that R is reflexive.Definition 10 (see [25]).
LetU,R,ψj be a PjS and ⊤Pj be a topology generated by j-adhesion neighborhoods. The j-adhesion lower, j-adhesion upper approximations, and j-adhesion accuracy of M⊆U are respectively given by(1)
LjM = ∪O∈⊤Pj:O⊆M = intPjM.(2)
UjM = ∩F∈ΓPj:M⊆F = clPjM.(3)
θjM = |LjM|/UjM, where UjM≠0.


## 2.4. Approximations and Topologies Generated by Differentj-Neighborhoods and Ideals

In [23], Hosny presented an idea depending on generating different topologies by using j-neighborhoods and ideals and studied some of their properties.Theorem 5 (see [23]).
ConsiderU,R,ψj as a NjS and ℐ as an ideal on U. Then, the family ⊤jℐ = {M⊆U: ∀h∈M, Njh − M∈ℐ} forms a topology on U for each j.Theorem 6 (see [23]).
LetU,R,ψj be a NjS and ℐ be an ideal on U. Then, ⊤j⊆⊤jℐ for each j.Definition 11 (see [23]).
LetU,R,ψj be a NjS and ℐ be an ideal on U. We called a set M⊆U an ℐj-open set if M∈⊤jℐ, and we called its complement an ℐj-closed set. The family Γjℐ of all ℐj-closed sets of a j-neighborhood space is given by Γjℐ = K⊆U:Kc∈⊤jℐ.Definition 12 (see [23]).
LetU,R,ψj be a NjS and ℐ be an ideal on U. The ℐj-lower, ℐj-upper approximations, and ℐj-accuracy of the approximation of M⊆U are respectively given by(1)
R¯jℐM = ∪O∈⊤jℐ:O⊆M = intjℐM, where intjℐM is an ℐj interior of M.(2)
R¯jℐM = ∩F∈Γjℐ:M⊆F = cljℐM, where cljℐM is an ℐj closure of M.(3)
αjℐM = |R¯jℐM|/R¯jℐM, where R¯jℐM≠0.


## 2.5. Approximations and Topologies Generated by DifferentEj-Neighborhoods and Ideals

For any binary relation, Hosny et al. [29] utilized the concepts of Ej-neighborhoods and ideal ℐ to output various topologies ζjℐ which are finer than the previous one generated by Ej-neighborhoods due to [26].Theorem 7 (see [29]).
The classζjℐ = {M⊆U: ∀h∈M, Ejh−M∈ℐ} forms a topology on U for each j.Definition 13 (see [29]).
LetU,R,ψj be a EjS and ℐ be an ideal on U. We called a set M⊆U a ζjℐ-open set if M∈ζjℐ, and we called its complement a ζjℐ-closed set. The class Πjℐ of all ζjℐ-closed sets is given by Πjℐ = K⊆U:Kc∈ζjℐ.Theorem 8 (see [29]).
ConsiderU,R,ψj as an EjS and ℐ as an ideal on U. Then,(1)
⊤Ej⊆ζjℐ.(2)
The reflexivity ofℛ implies that ζjℐ⊆ζjℐ for each j∈i,u,r,l.(3)
Ifℛ is a symmetric, then ζrℐ = ζlℐ = ζiℐ = ζuℐ and ζrℐ = ζlℐ = ζiℐ = ζuℐ.(4)
The transitivity ofℛ implies that ζjℐ⊆ζjℐ for each j∈i,u,r,l.(5)
Ifℛ is a preorder, then ζjℐ=ζjℐ for each j∈i,u,r,l.(6)
Ifℛ is an equivalence, then all ζjℐ are equal, and ⊤jℐ = ζjℐ for each j.Lemma 2 (see [29]).
For any binary relationℛ on U, we have Njh⊆Ejh for each h∈U and j∈i,u,r,l.Definition 14 (see [29]).
Considerζjℐ as a topology generated by Ej-neighborhoods and ideal ℐ. Then, for each j, ℐEj-lower, ℐEj-upper approximations, and ℐEj-accuracy of a subset M⊆U are defined respectively as follows:(1)
Ljℐ⊖M =  intEjℐM, where intEjℐM forms the interior points of M in ζjℐ.(2)
Ujℐ⊕M = clEjℐM, where clEjℐM forms the closure points of M in ζjℐ.(3)
σjℐM = |Ljℐ⊖M|/Ujℐ⊕M, where Ujℐ⊕M≠0.


## 3. Novel Topologies Generated fromj-Adhesion Neighborhoods via Ideals

Now, we deal with ideals and four differentj-adhesion neighborhoods generated from eight different j-neighborhoods via the same binary relation. By using them, new topologies are generated that generalize these topologies generated by j-adhesion neighborhoods. Several properties and relationships between these topologies are obtained.Throughout this paper, aPjS with ideal ℐU,R,ℐ,ψj is denoted by PjℐS.Theorem 9.
LetU,R,ℐ,ψj be a PjℐS. Then, the collection ρjℐ = {M⊆U: ∀h∈M, Pjh−M∈ℐ} forms a topology on U for each j.Proof.
First, letMα∈ρjℐ, α∈Δ, and z∈∪α∈ΔMα. Then, there is an α0∈Δ s.t. z∈Mα0. Therefore, Pjz−Mα0∈ℐ. Since −∪α∈ΔMα⊆−Mα0, Pjz−∪α∈ΔMα∈ℐ, i.e., ∪α∈ΔMα∈ρjℐ. Second, let M1,M2 be members of ρjℐ and z belong to the intersection of M1 and M2. Then, Pjz−M1∈ℐ and Pjz−M2∈ℐ. According to the definition of ℐ, we obtain Pjz−M1∪Pjz−M2∈ℐ. Hence, Pjz−M1∩M2∈ℐ. This means that M1∩M2∈ρjℐ. Finally, it is easy to see that ∅,U∈ρjℐ, ∀j∈r,l,i,u,r,l,i,u. Consequently, ρjℐ is a topology on U.Lemma 3.
Ifℐ,J are ideals on a PjSU,R,ψj such that ℐ is a subset of J, then ρjℐ⊆ρjJ.Proof.
Direct to prove.
The fundamental goal of the next results is to deduce the relations between the topologies generated byj-adhesion neighborhoods, topologies generated by j-neighborhoods and ideals, and topologies generated by j-adhesion neighborhoods and ideals, as it is shown in the following theorems.
Our new types of topologies, which were generated byj-adhesion neighborhoods and ideals, are finer than the previous one generated by j-adhesion neighborhoods due to [28] for any relation.Theorem 10.
LetU,R,ℐ,ψj be a PjℐS. If j∈r,l,i,u, then ⊤Pj⊆ρjℐ.Proof.
Straightforward.Example 1.
LetU = a,b,c,d and R = c,a,c,b,c,d,d,a,d,c,d,d. Then, we obtain the next topologies(1)⊤r=a,b,a,b,∅,U,⊤l=c,d,a,c,d,b,c,d,∅,U,⊤i=a,b,a,b,c,d,b,c,d,a,c,d,∅,U,⊤u=∅,U,⊤r=⊤u=a,d,a,b,d,a,c,d,∅,U,⊤l=⊤i=PU,⊤Pr=⊤Pl=c,d,a,b,c,d,a,b,d,a,b,c,∅,U,⊤Pl=⊤Pr=b,c,a,d,b,c,a,b,d,a,c,d,∅,U,⊤Pi=⊤Pi=PU,⊤Pu=⊤Pu=c,a,b,d,∅,U.
Ifℐ = ∅,a, then(2)⊤rℐ=a,b,a,b,b,c,d,∅,U,⊤lℐ=c,d,a,c,d,b,c,d,∅,U,⊤iℐ=a,b,c,d,a,b,b,c,d,a,c,d,∅,U,⊤uℐ=b,c,d,∅,U,⊤rℐ=⊤uℐ=d,a,d,b,d,c,d,a,b,d,a,c,d,b,c,d,∅,U,⊤lℐ=⊤lℐ=PU,ρrℐ=ρlℐ=b,c,d,a,b,b,c,c,d,b,d,a,b,c,a,b,d,b,c,d,∅,U,ρlℐ=ρrℐ=b,c,d,a,d,b,c,b,d,c,d,a,b,d,a,c,d,b,c,d,∅,U,ρiℐ=ρiℐ=PU,ρuℐ=ρuℐ=b,c,d,b,c,b,d,c,d,a,b,d,b,c,d,∅,U.
Ifℐ^ = ∅,d, then(3)⊤rℐ^=a,b,a,b,a,b,c,∅,U,⊤lℐ^=c,a,c,b,c,c,d,a,b,c,a,c,d,b,c,d,∅,U,⊤iℐ^=a,b,c,a,b,a,c,b,c,c,d,a,b,c,a,c,d,b,c,d,∅,U,⊤uℐ^=a,b,c,∅,U,⊤rℐ^=⊤uℐ^=a,a,b,a,c,a,d,a,b,c,a,b,d,a,c,d,∅,U,⊤lℐ^=⊤iℐ^==PU,ρrℐ^=ρlℐ^=c,d,a,b,c,d,a,b,c,a,b,d,∅,U,ρlℐ^=ρrℐ^=a,b,c,a,b,a,c,a,d,b,c,a,b,c,a,c,d,a,b,d,∅,U,ρiℐ^=ρiℐ^=PU,ρuℐ^=ρuℐ^=c,a,b,a,b,c,a,b,d,∅,U.
IfJ = ∅,a,d,a,d, then(4)⊤rJ=a,b,a,b,b,c,a,b,c,b,c,d,∅,U,⊤lJ=c,a,c,b,c,c,d,a,b,c,a,c,d,b,c,d,∅,U,⊤iJ=a,b,c,a,b,a,c,b,c,c,d,a,b,c,a,c,d,b,c,d,∅,U,⊤uJ=b,c,a,b,c,b,c,d,∅,U,⊤rJ=⊤uJ=⊤lJ=⊤iJ=PU,ρrJ=ρlJ=ρuJ=ρuJ=b,c,d,a,b,b,c,b,d,c,d,a,b,c,a,b,d,b,c,d,∅,U,ρlJ=ρrJ=ρiJ=ρiJ=PU.Remark 3.
Ifℐ = ∅,a in Example 1, then(1)
ρrℐ≠ρiℐ and ρlℐ≠ρiℐ.(2)
ρrℐ≠ρuℐ and ρlℐ≠ρuℐ.(3)
ρiℐ≠ρuℐ.Remark 4.
In view of Example1,(1)
In general, the collectionρjℐ need not be a quasidiscrete (clopen) topology on U, although ⊤Pj is quasidiscrete topology on U, ∀j∈i,i,r,r,l,l.(2)
If∀j∈u,u, then ρjℐ is not quasidiscrete topology on U.(3)
ρrℐ need not be dual topology to ρlℐ.(4)
IfR is any relation, then ⊤jℐ,ρjℐ are not comparable, for each j.(5)
Ifℐ = ∅ in Theorem 9, then the present generated topologies coincide with the previous one in Proposition 4.1 in [28]. So, the current work is considered as a generalization of work of [25, 27, 28].
Theorem11 illustrates that the reflexivity condition is necessary to create a relationship between the topologies ⊤jℐ and ρjℐ.Theorem 11.
LetU,R,ℐ,ψj be a PjℐS, where R is reflexive. Then, ⊤jℐ⊆ρjℐ, for any ideal ℐ.Proof.
Letℐ be any ideal on U. According to Proposition 3, ⊤jℐ⊆ρjℐ.Remark 5.
The new forms of topologies, which were generated byj-adhesion neighborhoods and ideals, are finer than the topologies generated by j-neighborhoods and ideals due to [23] for reflexive relation.Example 2.
LetU = a,b,c,d and R = ▲∪b,c,b,d,c,a,d,b,c,d,d,c.
Ifℐ = ∅,a, then(5)⊤rℐ=∅,U,a,b,c,d,⊤lℐ=∅,U,b,c,d,⊤iℐ=∅,U,a,b,c,d,⊤uℐ=∅,U,b,c,d,⊤rℐ=∅,U,a,c,d,a,c,d,b,c,d,⊤lℐ=∅,U,c,a,c,b,d,b,c,d,⊤iℐ=∅,U,a,c,d,a,c,a,d,b,d,c,d,a,b,d,a,c,d,b,c,d,⊤uℐ=∅,U,b,c,d,ρrℐ=ρlℐ=∅,U,a,c,a,c,b,d,a,b,d,b,c,d,ρlℐ=ρrℐ=∅,U,a,b,a,b,c,d,a,c,d,b,c,d,ρiℐ=ρiℐ=PU,ρuℐ=ρuℐ=∅,U,a,b,c,d.Theorem 12.
LetU,R,ℐ,ψj be a PjℐS and R be a symmetric relation on U. Then, ρrℐ = ρlℐ = ρiℐ = ρuℐ = ρrℐ = ρlℐ = ρiℐ = ρuℐ.Proof.
SinceR is a symmetric relation on U, then all j-adhesion neighborhoods coincide. So, ρrℐ = ρlℐ = ρiℐ = ρuℐ = ρrℐ = ρlℐ = ρiℐ = ρuℐ.Proposition 5.
LetU,R,ℐ,ψj be a PjℐS. Then, the following statements hold:(1)
ρrℐ = ρlℐ, ρlℐ = ρrℐ.(2)
ρiℐ = ρiℐ, ρuℐ = ρuℐ.(3)
ρuℐ⊆ρrℐ∩ρlℐ.(4)
ρrℐ∪ρlℐ⊆ρiℐ.(5)
ρuℐ⊆ρrℐ∩ρlℐ.(6)
ρrℐ∪ρlℐ⊆ρiℐ.Proof.
We prove (3) and one can prove the other cases in a similar way. LetM∈ρuℐ, then Puz−M∈ℐ, ∀z∈M. Hence, Prz∪Plz−M∈ℐ, ∀z∈M. So, Prz−M∈ℐ and Plz−M∈ℐ, ∀z∈M. Consequently, M∈ρrℐ∩ρlℐ.Definition 15.
LetU,R,ℐ,ψj be a PjℐS. A set M⊆U is called ρjℐ-open set if M∈ρjℐ, and its complement is called ρjℐ-closed set. The family ϒjℐ of all ρjℐ-closed sets is defined as ϒjℐ = K⊆U:Kc∈ρjℐ.Definition 16.
Considerρjℐ as a topology generated by j-adhesion neighborhoods and ideals. The ℐPj-lower, ℐPj-upper approximations, ℐPj-boundary regions, and ℐPj-accuracy of M⊆U are respectively given as follows:(1)
LjℐM = ∪O∈ρjℐ:O⊆M = intPjℐM, where intPjℐM represents interior of M wrt ρjℐ.(2)
UjℐM = ∩F∈ϒjℐ:M⊆F = clPjℐM, where clPjℐM represents closure of M wrt ρjℐ.(3)
BjℐM = UjℐM−LjℐM.(4)
θjℐM = |LjℐM|/UjℐM, where UjℐM≠0.Remark 6.
Table1 displays the comparison between j-approximations and j-accuracy for j=r depending on Definitions 10, 12, and 16 by using any relation R and ideal ℐ = ∅,a of Example 1.Table 1
Comparison betweenj-approximations and j-accuracy for each j=r depending on Definitions 10, 12, and 16 by using any relation R and ℐ = ∅,a of Example 1.
M⊆UOur method, Definition16Definition10Definition12LrℐMUrℐMθrℐMLrMUrMθrMR¯rℐMR¯rℐMαrℐMa∅a0∅a,b0aa1bba,b1/2∅a,b0bb,c,d1/3ccc1cc1∅c,d0ddd1dd1∅c,d0a,ba,ba,b1a,ba,b1a,bU1/2a,cca,c1/2ca,b,c1/3aa,c,d1/3a,dda,d1/2da,b,d1/3aa,c,d1/3b,cb,ca,b,c2/3ca,b,c1/3bb,c,d1/3b,db,da,b,d2/3da,b,d1/3bb,c,d1/3c,dc,dc,d1c,dc,d1∅c,d0a,b,ca,b,ca,b,c1a,b,ca,b,c1a,bU1/2a,b,da,b,da,b,d1a,b,da,b,d1a,bU1/2a,c,dc,da,c,d2/3c,dU1/2aa,c,d1/3b,c,db,c,dU3/4c,dU1/2b,c,db,c,d1UUU1UU1UU1Remark 7.
Table2 offers the comparison between j-approximations and j-accuracy for j=l depending on Definitions 12 and 16 by using a reflexive relation R and ideal ℐ = ∅,a of Example 2.
The next proposition that demonstrates the fundamental properties ofLjℐ, Ujℐ operators are understandable by observing that intPjℐ,clPjℐ fulfill all properties of the topological interior and closure operators, respectively.Table 2
Comparison betweenj-approximations and j-accuracy for each j=l depending on Definitions 12 and 16 by using a reflexive relation R and ℐ = ∅,a of Example 2.
M⊆UOur method, Definition16Definition12LlℐMUlℐMθlℐMR¯lℐMR¯lℐMαlℐMaaa1∅a0bbb1∅U0c∅c,d0∅U0d∅c,d0∅U0a,ba,ba,b1∅U0a,caa,c,d1/3∅U0a,daa,c,d1/3∅U0b,cbb,c,d1/3∅U0b,dbb,c,d1/3∅U0c,dc,dc,d1∅U0a,b,ca,bU1/2∅U0a,b,da,bU1/2∅U0a,c,da,c,da,c,d1∅U0b,c,db,c,db,c,d1b,c,dU3/4UUU1UU1Proposition 6.
LetU,R,ℐ,ψj be a PjℐS and j∈i,u,r,l. If M,M`⊆U, then the following conditions hold:L1LjℐM = UjℐMcc.U1UjℐM = LjℐMcc.L2LjℐU = U.U2Ujℐ∅ = ∅.L3 if M⊆M`, then LjℐM⊆LjℐM`.U3 if M⊆M`, then UjℐM⊆UjℐM`.L4LjℐM∩M` = LjℐM∩LjℐM`.U4UjℐM∪M` = UjℐM∪UjℐM`.L5LjℐM∪M`⊇LjℐM∪LjℐM`U5UjℐM∩M`⊆UjℐM∩UjℐM`.L6Ljℐ∅ = ∅.U6UjℐU = U.L7LjℐM⊆M.U7M⊆UjℐM.L8LjℐM = LjℐLjℐM.U8UjℐM = UjℐUjℐM.L9LjℐM⊆UjℐLjℐM.U9UjℐM⊇LjℐUjℐM.Remark 8.
Ifℐ = ∅,a and j=r, then Table 1 shows that the converse of L3,L5,L7,L9 does not hold.(1)
IfM = a, M` = b, then LrℐM⊆LrℐM`, but M⊈M`. Consequently, the converse of L3 is not true.(2)
IfM = a, M` = b, then LrℐM∪M` = a,b≠b = LrℐM∪LrℐM`. Consequently, the converse of L5 is not true.(3)
IfM = a,c, then the converse of L7 is not true.(4)
The equality of propertiesL9,U9 does not hold wrt the topology ρjℐ, although they are true according to the topology ⊤Pj. Here, if M = b, then LjℐM = b and UjℐLjℐM = a,b. Consequently, the converse of L9 does not hold.
By using any binary relation and without toting any supplement conditions as Proposition6 claims, one might observe that our approximation approach satisfies all features of the classical theory of rough set [1]. Thus, we can say that our presented method exemplifies a noteworthy generalization to RST.Proposition 7.
LetU,R,ℐ,ψj be a PjℐS and M⊆U. If j∈r,l,i,r,l,i, then the following conditions hold: LjℐM = M = UjℐM, ∀M∈℘U.Proposition 8.
LetU,R,ℐ,ψj be a PjℐS. If R is a reflexive relation on U, then LjℐM = UjℐLjℐM and UjℐM = LjℐUjℐM.
The following propositions are obvious and the proof is omitted.Proposition 9.
LetU,R,ℐ,ψj be a PjℐS and M⊆U. If j∈r,l,i,r,l,i, then the following statements hold:(1)
LuℐM⊆LrℐM⊆LiℐM and LuℐM⊆LlℐM⊆LiℐM.(2)
UiℐM⊆UrℐM⊆UuℐM and UiℐM⊆UlℐM⊆UuℐM.(3)
θuℐM≤θrℐM≤θiℐM and θuℐM≤θlℐM≤θiℐM.Proposition 10.
Letℐ,J be two ideals on a PjℐSU,R,ψj and M⊆U. If ℐ⊆J, then the following statements hold:(1)
LjℐM⊆LjJM.(2)
UjℐM⊇UjJM.(3)
θjℐM≤θjJM.
In Tables3 and 4, we make comparisons between j-approximations and j-accuracy for each j∈i,u,r,l.Table 3
Comparison betweenj-approximations and j-accuracy for each j∈i,u,r,l depending on Definition 16 by using any relation R and ℐ = ∅,a.
M⊆ULrℐMUrℐMθrℐMLlℐMUlℐMθlℐMLiℐMUiℐMθiℐMLuℐMUuℐMθuℐMa∅a0∅a0aa1∅a0bba,b1/2bb1bb1ba,b1/2ccc1cc1cc1cc1ddd1da,d1/2dd1da,d1/2a,ba,ba,b1ba,b1/2a,ba,b1ba,b1/2a,cca,c1/2ca,c1/2a,ca,c1ca,c1/2a,dda,d1/2a,da,d1a,da,d1da,d1/2b,cb,ca,b,c2/3b,cb,c1b,cb,c1b,ca,b,c2/3b,db,da,b,d2/3b,da,b,d2/3b,db,d1b,da,b,d2/3c,dc,dc,d1c,da,c,d2/3c,dc,d1c,da,c,d2/3a,b,ca,b,ca,b,c1b,ca,b,c2/3a,b,ca,b,c1b,ca,b,c2/3a,b,da,b,da,b,d1a,b,da,b,d1a,b,da,b,d1a,b,da,b,d1a,c,dc,da,c,d2/3a,c,da,c,d1a,c,da,c,d1c,da,c,d2/3b,c,db,c,dU3/4b,c,dU3/4b,c,db,c,d1b,c,dU3/4UUU1UU1UU1UU1Table 4
Comparison betweenj-approximations and j-accuracy for each j∈r,l,i,u depending on Definition 16 by using reflexive relation R and ℐ = ∅,a.
M⊆ULrℐMUrℐMθrℐMLlℐMUlℐMθlℐMLiℐMUiℐMθiℐMLuℐMUuℐMθuℐMaaa1aa1aa1aa1b∅b,d0bb1bb1∅b,c,d0ccc1∅c,d0cc1∅b,c,d0d∅b,d0∅c,d0dd1∅b,c,d0a,baa,b,d1/3a,ba,b1a,ba,b1aU1/4a,ca,ca,c1aa,c,d1/3a,ca,c1aU1/4a,daa,b,d1/3aa,c,d1/3a,da,d1aU1/4b,ccb,c,d1/3bb,c,d1/3b,cb,c1∅b,c,d0b,db,db,d1bb,c,d1/3b,db,d1∅b,c,d0c,dcb,c,d1/3c,dc,d1c,dc,d1∅b,c,d0a,b,ca,cU1/2a,bU1/2a,b,ca,b,c1aU1/4a,b,da,b,da,b,d1a,bU1/2a,b,da,b,d1aU1/4a,c,da,cU1/2a,c,da,c,d1a,c,da,c,d1aU1/4b,c,db,c,db,c,d1b,c,db,c,d1b,c,db,c,d1b,c,db,c,d1UUU1UU1UU1UU1Definition 17.
LetU,R,ℐ,ψj be a PjℐS and j∈i,i,r,r,l,l. A subset M of U is called(1)
Totallyℐj definable, if LjℐM = M = UjℐM.(2)
Internallyℐj definable, if LjℐM = M and UjℐM≠M.(3)
Externallyℐj definable, if LjℐM≠M and UjℐM = M.(4)
ℐj rough set, if LjℐM≠M and UjℐM≠M.Remark 9.
According to Table3, c is totally ℐr-definable, d is internally ℐl-definable, and a is externally ℐu-definable.


## 4. Comparison of Approximations and Accuracy Measures Generated byEj-Neighborhoods and Pj-Neighborhoods with Ideals

In this part, the comparison of approximations and accuracy measures generated byEj-neighborhoods and Pj-neighborhoods with ideals is discussed.In view of Example1, we conclude that ζjℐ and ρjℐ are not comparable, for each j, as we see in the next example.Example 3.
Continued from Example1, if ℐ = ∅,a, then(6)ζrℐ=ζiℐ=∅,U,a,b,a,b,c,d,a,c,d,b,c,d,ζlℐ=ζuℐ=∅,U,c,d,b,c,d,ζrℐ=ζuℐ=∅,U,b,c,d,ζlℐ=ζiℐ=PU.
According to Proposition3, our new sorts of topologies, which were generated by j-adhesion neighborhoods and ideal, are finer than the topologies generated by Ej-neighborhoods and ideals due to [29] for reflexive relation, as we see in the next theorem.Theorem 13.
LetR be any arbitrary binary relation on U. Then, for any ideal ℐ, the following statements are true:(1)
IfR is a reflexive relation on U, then ζjℐ⊆⊤Pj⊆ρjℐ, for each j.(2)
IfR is an equivalence relation on U, then ρjℐ = ⊤jℐ = ζjℐ, for each j.Proof.
Direct to prove.Proposition 11.
LetR (respectively ℐ) be a reflexive binary relation (respectively an ideal) on U and M⊆U. If j∈r,l,i,r,l,i, then the following statements hold:(1)
Ljℐ⊖M⊆LjM⊆LjℐM.(2)
UjℐM⊆UjM⊆Ujℐ⊕M.(3)
σjℐM≤θjM≤θjℐM.To uphold the acquired outcomes, we consider a reflexive relationR on U given in Example 2. If ℐ = ∅,a, then(7)ζrℐ=ζuℐ=ζuℐ=∅,U,b,c,d,ζlℐ=ζiℐ=ζrℐ=∅,U,a,b,c,d,ζlℐ=∅,U,c,a,c,b,d,b,c,d,ζiℐ=∅,U,a,c,a,c,b,d,a,b,d,b,c,d.Remark 10.
Table5 offers the comparison between j-approximations and j-accuracy for j=i depending on Definitions 14 and 16 by using a reflexive relation R and ideal ℐ = ∅,a of Example 2.Table 5
Comparison betweenj-approximations and j-accuracy for each j=i depending on Definitions 14 and 16 by using a reflexive relation R and ℐ = ∅,a of Example 2.
M⊆UOur method, Definition16Definition14LiℐMUiℐMθiℐMLiℐ⊖MUiℐ⊕MσiℐMaaa1aa1bbb1∅b,d0ccc1cc1ddd1∅b,d0a,ba,ba,b1aa,b,d1/3a,ca,ca,c1a,ca,c1a,da,da,d1aa,b,d1/3b,cb,cb,c1cb,c,d1/3b,db,db,d1b,db,d1c,dc,dc,d1cb,c,d1/3a,b,ca,b,ca,b,c1a,cU1/2a,b,da,b,da,b,d1a,b,da,b,d1a,c,da,c,da,c,d1a,cU1/2b,c,db,c,db,c,d1b,c,db,c,d1UUU1UU1


## 5. Conclusions and Future Works

To deal with uncertainty issues, Pawlak [1] proposed a nonstatistical approach called rough set. Its idea is based on collection of the elements that have the same values according the required attribute. Accuracy measures and approximations represent the essential ideas in rough set theory which give details of the boundary region in terms of size and structure.In this paper, we have suggested new techniques to generate new types of accuracy measures and approximations for a rough set under any arbitrary relation. These techniques are induced from hybridization of topological structures and ideals. In addition to studying their basic properties, we have compared between them. Then, we have compared them with last methods to show their importance to improve the approximations and maximize the accuracy measures.In the forthcoming articles, we are going to introduce further methods that help to obtain better approximation and higher accuracy measures than those given in the literature. Also, we search how these methods can be applied to model real-life issues.



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