# Topologies Generated by Two Ideals and the Correspondingj-Approximations Spaces with Applications

Authors: Mona Hosny
Journal: Journal of Mathematics
 (2021)

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## Abstract

Ideal is a fundamental concept in topological spaces and plays an important role in the study of topological problems. This motivated us to use two ideals to generate different topologies to take the advantage of the two ideals at the same time. Two ideals represent two opinions instead of one opinion which is very useful for using the insights of two groups of experts to study the problem and elicit decisions based on their common vision. Topology is a rich source for constructs that is helpful to enrich the original model of approximations spaces. Rough set theory has inbuilt topological concepts. Hence, the main purpose of this paper is to point out that the concept of rough sets has a purely topological aspects nature. To do so, new approximations spaces are introduced and defined based on the topologies generated by two ideals. The results in this paper show that the topological concepts can be a powerful method to study rough set models. The basic properties of these approximations are studied and compared to the previous ones and shown to be more general. The importance of the current paper is not only introducing a new kind of rough set based on bi-ideals, increasing the accuracy measure, and reducing the boundary region of the sets which is the main aim of rough set but also introducing a chemical application to explain the concepts.


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## Body

## 1. Introduction

It is well known that Pawlak’s rough set theory [1, 2] is constructed on the basis of an equivalence relation or a partition. An equivalence relation or a partition plays an important role in Pawlak’s rough set; however, it seems to be restrictive for many applications. Many generalizations to Pawlak space in order to change the constraints of the equivalence relation have been proposed [3–8]. Topology is a useful theoretic framework for the study of this theory as the topological notions are closely related to the notions and results of this theory. Lower and upper approximations operators are the central concepts in this theory and have the properties of the closure and interior operators. So, approximations operators in the rough sets theory are topological operators. Hence, the conjoint investigation of the rough set theory and topology becomes essential. Therefore, there are several generalizations of this theory, and each generalization employs topological concepts. The concept of the topological rough set by Wiweger [9] in 1989 is one of the most important topological generalizations of rough sets. The significant conjoint studies of the rough set theory and topology have been done by many researchers [9–23]. Ideal in topological spaces has been considered since 1930. This concept has won its importance by Vaidyanathaswamy [24]. In 1990, Jankovic and Hamlett [25] further introduced the ideal topological spaces and their applications to various fields. After the advent of the concept of ideals, there have been many great attempts, so far, by topologists to apply the notion of ideals for manoeuvring investigations of different problems of topology. So, many researchers [26–33] were interested in applying the concept of the ideals in the rough set theory. Abd El-Monsef et al. [34] introduced mixed neighborhood systems to approximate the rough sets. After that, Abd El-Monsef et al. [35] applied the concept of “j-neighborhood space” (briefly, j–NS) to generalize the classical rough set theory by using different general topologies induced from binary relations. In 2020, Hosny [28] generated different topologies by using the notion of ideals. Additionally, she defined the approximations by using these topologies as a generalization of [34, 35].This paper aims to study a certain extension problem via two ideals. Aware of the fact that ideals play an important role in the study of topological problems, this work is focused on generating different topologies by using two ideals instead of one. Moreover, the current study highlights the interdependencies of these topologies and the classical rough set theory. This paper comprises 6 sections and its sequence is as follows. Section2 outlines some definitions and basic concepts of topology and the rough set theory. Different topologies are induced by two ideals in Section 3. The comparisons between these topologies are introduced. The present topologies are more general than the previous ones in [28, 35]. It is shown that these topologies are finer than the previous ones. The current definitions coincide with Abd El-Monsef et al.’s [35] definitions if the two ideals are empty set and coincided with Hosny’s [28] definitions if the two ideals are equal to each other. So, the previous definitions [28, 35] are a special case of the current definitions. Section 4 contains two methods to propose new approximation spaces by using two ideals. In the first method, the lower approximations are defined by the union of two lower approximations, whereas the upper approximations are defined by the intersection of two upper approximations. The second one is based on the generated topologies which are introduced in the previous section. Moreover, the main properties of these methods are presented. These methods are an extension of the previous approximations in [28, 35]. The present methods reduced the boundary region and the current accuracy is greater than the previous ones. At the end of this section, the relationships among the current approximations are summarized in Table 1. It should be noted that even though Kandil et al. [31] presented two kinds of approximations via ideals in which one of them was based on one ideal and the other depended on two ideals, both of them depended on one neighborhood only. It is considered to be the main difference between these approximations and the current approximations which are based on open (closed) sets of topology (family of closed sets) generated by j-neighborhood and two ideals. Thus, there exist many differences between the main properties of the current approach and the previous one [31]. At the end of this work, an applied example is suggested in the chemistry field by using the current methods to illustrate the definitions in a friendly way. Section 6 compiles the conclusion of this work.Table 1
Comparison between theℐ1∨ℐ2-j-lower, ℐ1∨ℐ2-j-upper approximations, ℐ1∨ℐ2-j-boundary regions, and ℐ1∨ℐ2-j-accuracy is calculated by using the current approximations in Definition 14.
AThe current approximations in Definition14 for j=rThe current approximations in Definition14 for j=lThe current approximations in Definition14 for j=uThe current approximations in Definition14 for j=iR¯rℐ1∨ℐ2AR¯rℐ1∨ℐ2ABrℐ1∨ℐ2Aσrℐ1∨ℐ2AR¯lℐ1∨ℐ2AR¯lℐ1∨ℐ2ABlℐ1∨ℐ2Aσlℐ1∨ℐ2AR¯uℐ1∨ℐ2AR¯uℐ1∨ℐ2ABuℐ1∨ℐ2Aσuℐ1∨ℐ2AR¯iℐ1∨ℐ2AR¯iℐ1∨ℐ2ABiℐ1∨ℐ2Aσiℐ1∨ℐ2A111ϕ111,2,32,31/311,2,32,31/311ϕ1222ϕ1ϕ220ϕ22022ϕ13ϕ330ϕ330ϕ33033ϕ1443,431/244ϕ143,431/244ϕ11,21,21,2ϕ11,21,2,332/31,21,2,332/31,21,2ϕ11,311,331/21,31,2,322/311,2,32,31/31,31,3ϕ11,41,41,3,432/31,4U2,31/21,4U2,31/21,41,4ϕ12,322,331/2ϕ2,32,30ϕ2,32,302,32,3ϕ12,42,42,3,432/342,421/242,3,42,31/32,42,4ϕ13,43,43,4ϕ143,431/243,431/23,43,4ϕ11,2,31,21,2,332/31,2,31,2,3ϕ11,21,2,332/31,2,31,2,3ϕ11,2,41,2,4U33/41,2,4U33/41,2,4U33/41,2,41,2,4ϕ11,3,41,3,41,3,4ϕ11,3,4U23/41,3,4U23/41,3,41,3,4ϕ12,3,42,3,42,3,4ϕ142,3,42,31/342,3,42,31/32,3,42,3,4ϕ1UUUϕ1UUϕ1UUϕ1UUϕ1For example, takeA=1,3; then, Biℐ1∨ℐ2A=ϕ⊆3=Brℐ1∨ℐ2A⊆2,3=Buℐ1∨ℐ2A. Additionally, Biℐ1∨ℐ2A=ϕ⊆2=Blℐ1∨ℐ2A⊆2,3=Buℐ1∨ℐ2A.


## 2. Preliminaries

Definition 1.
(see [25]). A nonempty collection ℐ of subsets of a set U is called an ideal on U if it satisfies the following conditions:(1)
A∈ℐ and B∈ℐ⇒A∪B∈ℐ(2)
A∈ℐ and B⊆A⇒B∈ℐThat is,ℐ is closed under finite unions and subsets.Definition 2.
(see [31]). Let ℐ1,ℐ2 be two ideals on a nonempty set U. The smallest collection generating by ℐ1,ℐ2 is denoted by ℐ1∨ℐ2 and defined as(1)ℐ1∨ℐ2=G∪F:G∈ℐ1,F∈ℐ2.Proposition 1 (see [31]).
Ifℐ1,ℐ2 are two ideals on a nonempty set U and A,B are two subsets of U. Then, the collection ℐ1∨ℐ2 satisfies the following conditions:(1)
ℐ1∨ℐ2≠ϕ(2)
A∈ℐ1∨ℐ2,B⊆A⇒B∈ℐ1∨ℐ2(3)
A,B∈ℐ1∨ℐ2⇒A∪B∈ℐ1∨ℐ2
It means that the collectionℐ1∨ℐ2 is an ideal on U.Definition 3.
(see [35]). Let R be an arbitrary binary relation on a nonempty finite set U. The j-neighborhood of x∈UNjx,j∈r,l,r,l,i,u,i,u is defined as(1)
r-neighborhood: Nrx=y∈U:xRy(2)
l-neighborhood: Nlx=y∈U:yRx(3)
r-neighborhood: Nrx=∩x∈NryNry(4)
l-neighborhood: Nlx=∩x∈NlyNly(5)
i-neighborhood: Nix=Nrx∩Nlx(6)
u-neighborhood: Nux=Nrx∪Nlx(7)
i-neighborhood: Nix=Nrx∩Nlx(8)
u-neighborhood: Nux=Nrx∩NlxDefinition 4.
(see [35]). Let R be an arbitrary binary relation on a nonempty finite set U and let ξj:U⟶PU be a mapping which assigns for each x in U its j-neighborhood in PU. The triple U,R,ξj is called a j-neighborhood space (briefly, j–NS).Theorem 1 (see [35]).
LetU,R,ξj be a j–NS, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u, the collection τj=A⊆U:∀p∈A,Njp⊆A is a topology on U.Definition 5.
(see [35]). Let U,R,ξj be a j–NS. A subset A⊆U is called a j-open set if A∈τj, and the complement of a j-open set is called a j-closed set. The family Γj of all j-closed sets of a j-neighborhood space is defined by Γj=F⊆U:F′∈τj, where F′ is the complement of F.Definition 6.
(see [35]). Let U,R,ξj be a j–NS, A⊆U, and ∀j∈r,l,r,l,i,u,i,u. The j-lower, j-upper approximations, j-boundary regions, and j-accuracy of A are defined, respectively, as(2)R¯jA=∪G∈τj:G⊆A=intjA,whereintjArepresentsjinterior ofA,R¯jA=∩H∈Γj:A⊆H=cljA,wherecljArepresentsjclosure ofA,BjA=R¯jA−R¯jA,σjA=R¯jAR¯jA,whereR¯jA≠0.Definition 7.
(see [35]). Let U,R,ξj be a j–NS and ∀j∈r,l,r,l,i,u,i,u. A subset A⊆U is called a j-exact set if R¯jA=R¯jA. Otherwise, A is called a j-rough set.Theorem 2 (see [28]).
LetU,R,ξj be a j–NS, A⊆U, and ℐ an ideal on U. Then, ∀j∈r,l,r,l,i,u,i,u, and the collection τjℐ=A⊆U:∀p∈A,Njp∩A′∈ℐ is a topology on U.Theorem 3 (see [28]).
LetU,R,ξj be a j–NS, A⊆U, and ℐ an ideal on U. Then, ∀j∈r,l,r,l,i,u,i,u, τj⊆τjℐ.Definition 8.
(see [28]). Let U,R,ξj be a j–NS and ℐ an ideal on U. A subset A⊆U is called an ℐj-open set if A∈τjℐ and the complement of an ℐj-open set is called anℐj-closed set. The family Γjℐ of all ℐj-closed sets of a j-neighborhood space is defined by Γjℐ=F⊆U:F′∈τjℐ.Definition 9.
(see [28]). Let U,R,ξj be a j–NS, A⊆U,ℐ an ideal on U, and ∀j∈r,l,r,l,i,u,i,u. The ℐj-lower, ℐj-upper approximations, ℐj-boundary regions, and ℐj-accuracy of the approximations of A are defined, respectively, as(3)R¯jℐA=∪G∈τjℐ:G⊆A=intjℐA,whereintjℐArepresentsℐ−j−interior ofA,R¯jℐA=∩H∈Γjℐ:A⊆H=cljℐA,wherecljℐArepresentsℐ−j−closure ofA,BjℐA=R¯jℐA−R¯jℐA,σjℐA=R¯jℐAR¯jℐA,whereR¯jℐA≠0.Definition 10.
(see [28]). Let U,R,ξj be a j–NS, ℐ an ideal on U,andA⊆U,∀j∈r,l,r,l,i,u,i,u. A subset A is called an ℐ-j-definable (ℐ-j-exact) set if R¯jℐA=R¯jℐA. Otherwise, A is called an ℐ-j-rough set.Corollary 1 (see [28]).
LetU,R,ξj be a j–NS, ℐ an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u.(1)
Everyj-exact subset in U is ℐ-j-exact(2)
Everyℐ-j-rough subset in U is j-rough


## 3. Topology Based on Different Neighborhoods by Using Bi-Ideals

In this section, topologies based on different neighborhoods are generated by using bi-ideals as a generalization of the previous results in [28,35]. The main properties of these topologies are studied and compared to the previous ones in [28,35] and shown to be more general.Theorem 4.
LetU,R,ξj be a j–NS, A⊆U, and ℐ1∨ℐ2 an ideal on U. Then, ∀j∈r,l,r,l,i,u,i,u, and the collection τjℐ1∨ℐ2=A⊆U:∀p∈A,Njp∩A′∈ℐ1∨ℐ2 is a topology on U.Proof.
(1)
Clearly,U and ϕ belong to τjℐ1∨ℐ2.(2)
LetAδ∈τjℐ1∨ℐ2∀δ∈Δ and a∈∪δ∈ΔAδ. Then, ∃γ0∈Δ such that a∈Aδ0:(4)⇒Nja∩Aδ0′∈ℐ1∨ℐ2⇒Nja∩∪i∈IAδ′∈ℐ1∨ℐ2⇒∪δ∈ΔAδ∈τjℐ1∨ℐ2.(3)
LetA,B∈τjℐ1∨ℐ2 and a∈A∩B:(5)⇒Nja∩A′∈ℐ1∨ℐ2andNja∩B′∈ℐ1∨ℐ2⇒Nja∩A′∪Nja∩B′∈ℐ1∨ℐ2⇒Nja∩A′∪B′∈ℐ1∨ℐ2⇒Nja∩A∩B′∈ℐ1∨ℐ2⇒A∩B∈τjℐ1∨ℐ2.
From 1, 2, and 3,τjℐ1∨ℐ2 is a topology on U.
The following theorem shows that the present topologies are than the previous ones in [28, 35].Theorem 5.
LetU,R,ξj be a j–NS, A⊆U, and ℐ1∨ℐ2 an ideal on U. Then, ∀j∈r,l,r,l,i,u,i,u, τj⊆τjℐk⊆τjℐ1∨ℐ2,k∈1,2.Proof.
τj⊆τjℐk by Theorem 3 [28]. To prove τjℐk⊆τjℐ1∨ℐ2, let A∈τjℐk. Then, Njp∩A′∈ℐk,∀p∈A. Therefore, Njp∩A′∈ℐ1∨ℐ2,∀p∈A. So, A∈τjℐ1∨ℐ2. Hence, τjℐk⊆τjℐ1∨ℐ2.Remark 1.
The following points should be noted:(1)
Ifℐ1∨ℐ2=ϕ in Theorem 5, then the current generated topologies coincide with the previous one in Theorem 1 [35]. So, the present results are considered as a generalization of Abd El-Monsef et al.’s work [35].(2)
Ifℐ1=ℐ2 in Theorem 5, then the current generated topologies coincide with the previous one in Theorem 2 [28]. So, the present results are an extension of Hosny’s work [28].(3)
τj⊊τjℐk⊊τjℐ1∨ℐ2 as shown in the following example.Example 1.
LetU=1,2,3,4,R=1,1,1,2,1,3,2,3,3,4, ℐ1=ϕ,2,ℐ2=ϕ,3, and ℐ1∨ℐ2=ϕ,2,3,2,3. The following results are clear: (1)
τrℐ1=U,ϕ,4,3,4,1,3,4,2,3,4,τrℐ2=U,ϕ,2,4,1,2,2,4,3,4,1,2,4,2,3,4, and τrℐ1∨ℐ2=U,ϕ,1,2,4,1,2,1,4,2,4,3,4,1,2,4,1,3,4,2,3,4. Thus, τrℐ1,τrℐ2⊊τrℐ1∨ℐ2.(2)
τlℐ1=U,ϕ,1,1,2,1,3,1,2,3,1,3,4,τlℐ2=U,ϕ,1,4,1,2,1,4,1,2,3,1,2,4, and τlℐ1∨ℐ2=U,ϕ,1,4,1,2,1,3,1,4,1,2,3,1,2,4,1,3,4. Thus, τlℐ1,τlℐ2⊊τlℐ1∨ℐ2.(3)
τiℐ1=PU−1,4,τiℐ2=τiℐ1∨ℐ2=PU. Thus, τiℐ1⊊τiℐ1∨ℐ2.(4)
τuℐ1=U,ϕ,1,3,4,τuℐ2=U,ϕ,4,1,2,1,2,4, and τuℐ1∨ℐ2=U,ϕ,1,4,1,2,1,4,1,2,4,1,3,4. Thus, τuℐ1,τuℐ2⊊τuℐ1∨ℐ2.(5)
τrℐ1=U,ϕ,3,4,1,3,3,4,1,2,3,1,3,4,τrℐ2=U,ϕ,3,4,1,2,3,4,1,2,3,1,2,4, and τrℐ1∨ℐ2=U,ϕ,1,3,4,1,2,1,3,1,4,3,4,1,2,3,1,2,4,1,3,4. Thus, τrℐ1,τrℐ2⊊τrℐ1∨ℐ2.(6)
τuℐ1=U,ϕ,3,4,1,3,3,4,1,2,3,1,3,4,τuℐ2=U,ϕ,3,4,1,2,3,4,1,2,3,1,2,4, and τuℐ1∨ℐ2=U,ϕ,1,3,4,1,2,1,3,1,4,3,4,1,2,3,1,2,4,1,3,4. Thus, τuℐ1,τuℐ2⊊τuℐ1∨ℐ2.
Similarly, we can add an example to show that(1)
τlℐk⊊τlℐ1∨ℐ2(2)
τiℐk⊊τiℐ1∨ℐ2Proposition 2.
LetU,R,ξj be a j–NS and ℐ1∨ℐ2 an ideal on U. Then,(1)
τuℐ1∨ℐ2⊆τrℐ1∨ℐ2 and τuℐ1∨ℐ2⊆τlℐ1∨ℐ2(2)
τrℐ1∨ℐ2⊆τiℐ1∨ℐ2 and τlℐ1∨ℐ2⊆τiℐ1∨ℐ2(3)
τuℐ1∨ℐ2⊆τrℐ1∨ℐ2 and τuℐ⊆τlℐ1∨ℐ2(4)
τrℐ1∨ℐ2⊆τiℐ1∨ℐ2 and τlℐ1∨ℐ2⊆τiℐ1∨ℐ2Proof.
(1)
LetA∈τuℐ1∨ℐ2. Then, Nup∩A′∈ℐ1∨ℐ2,∀p∈A. Thus, Nrp∪Nlp∩A′∈ℐ1∨ℐ2,∀p∈A. Hence, Nrp∩A′∈ℐ1∨ℐ2,∀p∈A and Nlp∩A′∈ℐ1∨ℐ2,∀p∈A. Therefore, A∈τrℐ1∨ℐ2 and A∈τlℐ1∨ℐ2. Hence, τuℐ1∨ℐ2⊆τrℐ1∨ℐ2 and τuℐ1∨ℐ2⊆τlℐ1∨ℐ2. Similarly, we can prove 3.(2)
LetA∈τrℐ1∨ℐ2. Then, Nrp∩A′∈ℐ1∨ℐ2,∀p∈A. Thus, Nrp∩Nlp∩A′∈ℐ1∨ℐ2,∀p∈A. Hence, Nip∩A′∈ℐ1∨ℐ2,∀p∈A. Therefore, A∈τiℐ1∨ℐ2. Hence, τrℐ1∨ℐ2⊆τiℐ1∨ℐ2. Similarly, we can prove 4.Corollary 2.
LetU,R,ξj be a j–NS and ℐ1∨ℐ2 an ideal on U. Then, ∀j∈r,l,r,l,i,u,i,u.(1)
τuℐ1∨ℐ2⊆τrℐ1∨ℐ2⊆τiℐ1∨ℐ2(2)
τuℐ1∨ℐ2⊆τlℐ1∨ℐ2⊆τiℐ1∨ℐ2(3)
τuℐ1∨ℐ2⊆τrℐ1∨ℐ2⊆τiℐ1∨ℐ2(4)
τuℐ1∨ℐ2⊆τlℐ1∨ℐ2⊆τiℐ1∨ℐ2Remark 2.
(i)
Example1 shows that the inclusion in Proposition 2 and Corollary 2 cannot be replaced by equality relation as follows:(1)
τrℐ1∨ℐ2⊈τuℐ1∨ℐ2 and τiℐ1∨ℐ2⊈τrℐ1∨ℐ2(2)
τlℐ1∨ℐ2⊈τuℐ1∨ℐ2 and τiℐ1∨ℐ2⊈τlℐ1∨ℐ2(ii)
Similarly, we can add an example to show that(1)
τrℐ1∨ℐ2⊈τuℐ1∨ℐ2 and τiℐ1∨ℐ2⊈τrℐ1∨ℐ2(2)
τlℐ1∨ℐ2⊈τuℐ1∨ℐ2 and τiℐ1∨ℐ2⊈τlℐ1∨ℐ2Remark 3.
LetU,R,ξj be a j–NS and ℐ1∨ℐ2 an ideal on U. Then, the following statements should be noted: (1)
τrℐ1∨ℐ2 is not the dual of τlℐ1∨ℐ2 (see Example 1). Although τr is the dual of τl, it is proved in [35].(2)
τrℐ1∨ℐ2 and τrℐ1∨ℐ2 are not necessarily to be comparable (see Example 1).(3)
τlℐ1∨ℐ2 and τlℐ1∨ℐ2 are not necessarily to be comparable (see Example 1).(4)
τiℐ1∨ℐ2 and τiℐ1∨ℐ2 are not necessarily to be comparable.(5)
IfR is a reflexive relation, then the following topologies are comparable as follows:(a)
τrℐ1∨ℐ2⊆τrℐ1∨ℐ2(b)
τlℐ1∨ℐ2⊆τlℐ1∨ℐ2(c)
τiℐ1∨ℐ2⊆τiℐ1∨ℐ2Definition 11.
LetU,R,ξj be a j–NS and ℐ1∨ℐ2 an ideal on U. A subset A⊆U is called an ℐ1∨ℐ2j-open set if A∈τjℐ1∨ℐ2 and the complement of an ℐ1∨ℐ2j-open set is called an ℐ1∨ℐ2j-closed set. The family Γjℐ1∨ℐ2 of all ℐ1∨ℐ2j-closed sets of a j-neighborhood space is defined by Γjℐ1∨ℐ2=F⊆U:F′∈τjℐ1∨ℐ2.Theorem 6.
LetU,R,ξj be a j–NS and ℐ1∨ℐ2,J1∨J2 two ideals on U. If ℐ1∨ℐ2⊆J1∨J2, then ∀j∈r,l,r,l,i,u,i,u,(1)
τjℐ1∨ℐ2⊆τjJ1∨J2(2)
Γjℐ1∨ℐ2⊆ΓjJ1∨J2Proof.
(1)
LetA∈τjℐ1∨ℐ2; then, ∀p∈A,Njp∩A′∈ℐ1∨ℐ2⊆J1∨J2. Thus, ∀p∈A,Njp∩A′∈J1∨J2. Hence, A∈τjJ1∨J2. Therefore, τjℐ1∨ℐ2⊆τjJ1∨J2.(2)
Immediately from (1).


## 4.ℐ1∨ℐ2-j-Approximations Spaces

In this section, more general notions of approximations are suggested. One of these notions is based on the union of two lower approximations and the intersections of the two upper approximations. On the other hand, there exists a close relationship between topologies and rough sets. So, the other notions explore rough set theory from the point of view of topology.Definition 12.
LetU,R,ξj be a j–NS, A⊆U,ℐk,∀k∈1,2 two ideals on U, and ∀j∈r,l,r,l,i,u,i,u. The ℐ1,ℐ2-j-lower, ℐ1,ℐ2-j-upper approximations, ℐ1,ℐ2-j-boundary regions, and ℐ1,ℐ2-j-accuracy of the approximations of A are defined, respectively, asR¯jℐ1,ℐ2A=R¯jℐ1A∪R¯jℐ2A, where R¯jℐkA is the ℐk-j-lower approximations of A with respect to ℐk,k∈1,2 as in Definition 9R¯jℐ1,ℐ2A=R¯jℐ1A∩R¯jℐ2A, where R¯jℐkA is the ℐk-j-upper approximations of A with respect to ℐk,k∈1,2 as in Definition 9Bjℐ1,ℐ2A=R¯jℐ1,ℐ2A−R¯jℐ1,ℐ2Aσjℐ1,ℐ2A=R¯jℐ1,ℐ2A/R¯jℐ1,ℐ2A, where R¯jℐ1,ℐ2A≠0Proposition 3.
LetU,R,ξj be a j–NS, A,B⊆U,ℐ1,ℐ2 two ideals on U, and ∀j∈r,l,r,l,i,u,i,u. Then, the following properties hold:(1)
R¯jℐ1,ℐ2A⊆A⊆R¯jℐ1,ℐ2AR¯jℐ1,ℐ2A⊆A⊆R¯jℐ1,ℐ2A; equality holds if A=ϕ or U(2)
A⊆B⇒R¯jℐ1,ℐ2A⊆R¯jℐ1,ℐ2BandR¯jℐ1,ℐ2A⊆R¯jℐ1,ℐ2B(3)
R¯jℐ1,ℐ2A∩B⊆R¯jℐ1,ℐ2A∩R¯jℐ1,ℐ2B(4)
R¯jℐ1,ℐ2A∪B⊇R¯jℐ1,ℐ2A∪R¯jℐ1,ℐ2B(5)
R¯jℐ1,ℐ2A∩B⊆R¯jℐ1,ℐ2A∩R¯jℐ1,ℐ2B(6)
R¯jℐ1,ℐ2A∪B⊇R¯jℐ1,ℐ2A∪R¯jℐ1,ℐ2B(7)
R¯jℐ1,ℐ2A=R¯jℐ1,ℐ2A′′,R¯jℐ1,ℐ2A=R¯jℐ1,ℐ2A′′Proof.
The proof is straightforward by using Definitions12 and 9 [28].Remark 4.
Example1 shows that(1)
The inclusion in parts 1, 3, 4, 5, and 6 of Proposition3 cannot be replaced by equality relation:(i)
For part 1, ifA=1,4, then R¯rℐ1,ℐ2A=4≠1,4=A≠1,3,4=R¯rℐ1,ℐ2A(ii)
For part 3, ifA=1,3,B=1,4, then R¯rℐ1,ℐ2A∩B=1≠1,3=R¯rℐ1,ℐ2A∩R¯rℐ1,ℐ2B(iii)
For part 4, ifA=2,B=3, then R¯rℐ1,ℐ2A∪B=1,2,3≠2,3=R¯rℐ1,ℐ2A∪R¯rℐ1,ℐ2B(iv)
For part 5, ifA=1,2,4,B=1,3,4, then R¯rℐ1,ℐ2A∩B=4≠1,4=R¯rℐ1,ℐ2A∩R¯rℐ1,ℐ2B(v)
For part 6, ifA=1,3,B=1,4, then R¯rℐ1,ℐ2A∪B=1,3,4≠4=R¯rℐ1,ℐ2A∪R¯rℐ1,ℐ2B(2)
The converse of part 2 of Proposition3 is not necessarily true; if A=3,B=4, then R¯rℐ1,ℐ2A=3⊆3,4=R¯rℐ1,ℐ2BandR¯rℐ1,ℐ2A=ϕ⊆4=R¯rℐ1,ℐ2B, but A⊈BDefinition 13.
LetU,R,ξj be a j–NS and ℐ1,ℐ2 two ideals on U,A⊆U,∀j∈r,l,r,l,i,u,i,u. A subset A is called an ℐ1,ℐ2-j-definable (an ℐ1,ℐ2-j-exact) set if R¯jℐ1,ℐ2A=R¯jℐ1,ℐ2A. Otherwise, A is called an ℐ1,ℐ2-j-rough set.Relationships between Abd El-Monsef et. al's approximations in Definition6 [28], Hosny's approximations in Definition 9 [35], and the present approximations in Definition 12 are given in the following proposition.Proposition 4.
LetU,R,ξj be a j–NS, A⊆U,ℐk,∀k∈1,2 two ideals on U, and ∀j∈r,l,r,l,i,u,i,u. Then,(1)
R¯jA⊆R¯jℐkA⊆R¯jℐ1,ℐ2A(2)
R¯jℐ1,ℐ2A⊆R¯jℐkA⊆R¯jA(3)
Bjℐ1,ℐ2A⊆BjℐkA⊆BjA(4)
σjA≤σjℐkA≤σjℐ1,ℐ2AProof.
Immediately by using Definitions12, 6 [35] and 9 [28].Definition 14.
LetU,R,ξj be a j–NS, A⊆U,ℐ1∨ℐ2 an ideal on U, and ∀j∈r,l,r,l,i,u,i,u. The ℐ1∨ℐ2-j-lower, ℐ1∨ℐ2-j-upper approximations, ℐ1∨ℐ2-j-boundary regions, and ℐ1∨ℐ2-j-accuracy of the approximations of A are defined, respectively, as(6)R¯jℐ1∨ℐ2A=∪G∈τjℐ1∨ℐ2:G⊆A=intjℐ1∨ℐ2A,whereintjℐ1∨ℐ2Arepresentsℐ1∨ℐ2−j−interior ofA,R¯jℐ1∨ℐ2A=∩H∈Γjℐ1∨ℐ2:A⊆H=cljℐ1∨ℐ2A,wherecljℐ1∨ℐ2Arepresentsℐ1∨ℐ2−j−interior ofA,Bjℐ1∨ℐ2A=R¯jℐ1∨ℐ2A−R¯jℐ1∨ℐ2A,σjℐ1∨ℐ2A=R¯jℐ1∨ℐ2AR¯jℐ1∨ℐ2A,whereR¯jℐ1∨ℐ2A≠0.The basic properties of the currentℐ1∨ℐ2-j-lower and ℐ1∨ℐ2-j-upper approximations are introduced in the following proposition.Proposition 5.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A,B⊆U. Then, ∀j∈r,l,r,l,i,u,i,u:(1)
R¯jℐ1∨ℐ2A⊆A⊆R¯jℐ1∨ℐ2A equality holds if A=ϕ or U(2)
A⊆B⇒R¯jℐ1∨ℐ2A⊆R¯jℐ1∨ℐ2B(3)
A⊆B⇒R¯jℐ1∨ℐ2A⊆R¯jℐ1∨ℐ2B(4)
R¯jℐ1∨ℐ2A∩B⊆R¯jℐ1∨ℐ2A∩R¯jℐ1∨ℐ2B(5)
R¯jℐ1∨ℐ2A∪B⊇R¯jℐ1∨ℐ2A∪R¯jℐ1∨ℐ2B(6)
R¯jℐ1∨ℐ2A∪B=R¯jℐ1∨ℐ2A∪R¯jℐ1∨ℐ2B(7)
R¯jℐ1∨ℐ2A∩B=R¯jℐ1∨ℐ2A∩R¯jℐ1∨ℐ2B(8)
R¯jℐ1∨ℐ2A=R¯jℐ1∨ℐ2A′′, R¯jℐ1∨ℐ2A=R¯jℐ1∨ℐ2A′′(9)
R¯jℐ1∨ℐ2R¯jℐ1∨ℐ2A=R¯jℐ1∨ℐ2A(10)
R¯jℐ1∨ℐ2R¯jℐ1∨ℐ2A=R¯jℐ1∨ℐ2A(11)
R¯jℐ1∨ℐ2R¯jℐ1∨ℐ2A⊆R¯jℐ1∨ℐ2R¯jℐ1∨ℐ2A(12)
R¯jℐ1∨ℐ2R¯jℐ1∨ℐ2A⊆R¯jℐ1∨ℐ2R¯jℐ1∨ℐ2AThe proof of this proposition is simple using the properties ofℐ1∨ℐ2-j-interior and ℐ1∨ℐ2-j-closure, so we omit it.Remark 5.
Example1 shows the following:(1)
The inclusion in parts 1, 4, 5, 11, and 12 of Proposition5 cannot be replaced by equality relation (take j=r):(i)
For part 1, ifA=3,R¯rℐ1∨ℐ2A=ϕ, then A⊈R¯rℐ1∨ℐ2A. If A=4,R¯rℐ1∨ℐ2A=3,4, then R¯rℐ1∨ℐ2A⊈A.(ii)
For part 4, ifA=3,B=4,A∩B=ϕ,R¯rℐ1∨ℐ2A=3,R¯rℐ1∨ℐ2B=3,4,R¯rℐ1∨ℐ2A∩B=ϕ, then R¯rℐ1∨ℐ2A∩R¯rℐ1∨ℐ2B=3⊈ϕ=R¯rℐ1∨ℐ2A∩B.(iii)
For part 5, ifA=3,B=4,A∪B=3,4,R¯rℐ1∨ℐ2A=ϕ,R¯rℐ1∨ℐ2B=4,R¯rℐ1∨ℐ2A∪B=3,4, then R¯rℐ1∨ℐ2A∪B=3,4⊈4=R¯rℐ1∨ℐ2A∪R¯rℐ1∨ℐ2B.(iv)
For part 11, ifA=4,R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A=A,R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A=3,4, then R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A⊈R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A.(v)
For part 12, ifA=3,R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A=A,R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A=ϕ, then R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A⊈R¯rℐ1∨ℐ2R¯rℐ1∨ℐ2A.(2)
The converse of parts 2 and 3 of Proposition5 is not necessarily true (take j=r):(i)
For part 2, ifA=3,B=4, then R¯rℐ1∨ℐ2A=3,R¯rℐ1∨ℐ2B=3,4. Therefore, R¯rℐ1∨ℐ2A⊆R¯rℐ1∨ℐ2B, but A⊈B.(ii)
For part 3, ifA=3,B=2, then R¯rℐ1∨ℐ2A=ϕ,R¯rℐ1∨ℐ2B=2. Therefore, R¯rℐ1∨ℐ2A⊆R¯rℐ1∨ℐ2B, but A⊈B.Definition 15.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U,A⊆U,and∀j∈r,l,r,l,i,u,i,u. A subset A is called an ℐ1∨ℐ2-j-definable (an ℐ1∨ℐ2-j-exact) set if R¯jℐ1∨ℐ2A=R¯jℐ1∨ℐ2A. Otherwise, A is called an ℐ1∨ℐ2-j-rough set.In Example1, A=2 is ℐ1∨ℐ2-r-exact, while B=3 is ℐ1∨ℐ2-r-rough.Remark 6.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u, the intersection of two ℐ1∨ℐ2-j-rough sets need not be an ℐ1∨ℐ2-j-rough set as in Example 1, 1,3 and 1,4 are ℐ1∨ℐ2-r-rough sets, and 1,3∩1,4=1 is not an ℐ1∨ℐ2-r-rough set.The following theorem and corollary present the comparisons between the present approximations in Definitions12 and 14 and the previous ones in Definitions 6 [35] and 9 [28].Theorem 7.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u and ∀k∈1,2.(1)
R¯jA⊆R¯jℐkA⊆R¯jℐ1,ℐ2A⊆R¯jℐ1∨ℐ2A(2)
R¯jℐ1∨ℐ2A⊆R¯jℐ1,ℐ2A⊆R¯jℐkA⊆R¯jAProof.
(1)
R¯jA⊆R¯jℐkA⊆R¯jℐ1,ℐ2 by Proposition 4. To prove, R¯jℐ1,ℐ2A⊆R¯jℐ1∨ℐ2A. Since R¯jℐ1,ℐ2A=R¯jℐ1A∪R¯jℐ2A and since R¯jℐkA=∪G∈τjℐk:G⊆A⊆∪G∈τjℐ1∨ℐ2:G⊆A=R¯jℐ1∨ℐ2A (by Theorem 5), R¯jℐ1,ℐ2A⊆R¯jℐ1∨ℐ2A.(2)
It is similar to (1).Corollary 3.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u and ∀k∈1,2.(1)
Bjℐ1∨ℐ2A⊆Bjℐ1,ℐ2A⊆BjℐkA⊆BjA(2)
σjA≤σjℐkA≤σjℐ1,ℐ2A≤σjℐ1∨ℐ2ACorollary 4.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u and ∀k∈1,2.(1)
A is j-exact ⇒A is ℐk-j-exact ⇒A is ℐ1,ℐ2-j-exact ⇒A is ℐ1∨ℐ2-j-exact(2)
A is ℐ1∨ℐ2-j-rough ⇒A is ℐ1,ℐ2-j-rough ⇒A is ℐk-j-rough ⇒A is j-roughTable 2
Comparison between the boundary and accuracy by using the current approximations in Definitions12 and 14 and the previous ones in Definitions 6 [35] AND 9 [28] at j=r and k=1.
AThe previous one in Definition6 [35]The previous one in Definition9 [28]The current method in Definition12The current method in Definition14R¯rAR¯rABrAσrAR¯rℐ1AR¯rℐ1ABrℐ1Aσrℐ1AR¯rℐ1,ℐ2AR¯rℐ1,ℐ2ABrℐ1,ℐ2Aσrℐ1,ℐ2AR¯rℐ1∨ℐ2AR¯rℐ1∨ℐ2ABrℐ1∨ℐ2Aσrℐ1∨ℐ2A1ϕ110ϕ110ϕ11011ϕ12ϕ1,21,20ϕ22022ϕ122ϕ13ϕ1,2,31,2,30ϕ1,2,31,2,30ϕ330ϕ33044U1,2,31/44U1,2,31/443,431/243,431/21,2ϕ1,21,20ϕ1,21,201,21,2ϕ11,21,2ϕ11,3ϕ1,2,31,2,30ϕ1,2,31,2,30ϕ1,31,3011,331/21,44U1,2,31/34U1,2,31/441,3,41,31/31,41,3,432/32,3ϕ1,2,31,2,30ϕ1,2,31,2,3021,2,31,31/322,331/22,44U1,2,31/44U1,2,31/42,4U1,31/22,42,3,432/33,43,4U1,21/23,4U1,21/23,43,4ϕ13,43,4ϕ11,2,3ϕ1,2,31,2,30ϕ1,2,31,2,301,21,2,332/31,21,2,332/31,2,44U1,2,31/44U1,2,31/41,2,4U33/41,2,4U33/41,3,43,4U1,21/21,3,4U23/41,3,41,3,4ϕ11,3,41,3,4ϕ12,3,42,3,4U13/42,3,4U13/42,3,4U13/42,3,42,3,4ϕ1UUUϕ1UUϕ1UUϕ1UUϕ1For example, takeA=1,2; then, the boundary and accuracy by the present method in Definitions 12 and 14 are ϕ and 1, respectively, whereas the boundary and accuracy by using Abd El-Monsef et al.’s method 6 [35] and Hosny’s 9 [28] are 1,2 and 0, respectively.Remark 7.
(1)
The converse of parts of Theorem7 and Corollaries 3 and 4 is not necessarily to be true as shown in Table 2. This table is calculated by using Example 1.(2)
Theorem7 shows that the present method in Definitions 12 and 14 reduces the boundary region by increasing the lower approximations and decreasing the upper approximations with the comparison of Abd El-Monsef et al.’s Definition 6 [35] and Hosny’s Definition 9 [28]. Moreover, Corollary 3 shows that the current accuracy in Definitions 12 and 14 is greater than the previous ones in Definitions 6 and 9.The following propositions and corollaries are studied the relationships among theℐ1∨ℐ2-j-lower, ℐ1∨ℐ2-j-upper approximations, ℐ1∨ℐ2-j-boundary regions, and ℐ1∨ℐ2-j-accuracy.Proposition 6.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u. Then, the following statements are true in general:(1)
R¯uℐ1∨ℐ2A⊆R¯rℐ1∨ℐ2A⊆R¯iℐ1∨ℐ2A(2)
R¯uℐ1∨ℐ2A⊆R¯lℐ1∨ℐ2A⊆R¯iℐ1∨ℐ2A(3)
R¯uℐ1∨ℐ2A⊆R¯rℐ1∨ℐ2A⊆R¯iℐA(4)
R¯uℐ1∨ℐ2A⊆R¯lℐ1∨ℐ2A⊆R¯iℐ1∨ℐ2AProof.
By using Proposition2, the proof is obvious.Proposition 7.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u. Then, the following statements are true in general:(1)
R¯iℐ1∨ℐ2A⊆R¯rℐ1∨ℐ2A⊆R¯uℐ1∨ℐ2A(2)
R¯iℐ1∨ℐ2A⊆R¯lℐA⊆R¯uℐA(3)
R¯iℐ1∨ℐ2A⊆R¯rℐ1∨ℐ2A⊆R¯uℐ1∨ℐ2A(4)
R¯iℐ1∨ℐ2A⊆R¯lℐ1∨ℐ2A⊆R¯uℐ1∨ℐ2AProof.
By using Proposition2, the proof is obvious.Corollary 5.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u.(1)
Biℐ1∨ℐ2A⊆Brℐ1∨ℐ2A⊆Buℐ1∨ℐ2A(2)
Biℐ1∨ℐ2A⊆Blℐ1∨ℐ2A⊆Buℐ1∨ℐ2A(3)
Biℐ1∨ℐ2A⊆Brℐ1∨ℐ2A⊆Buℐ1∨ℐ2A(4)
Biℐ1∨ℐ2A⊆Blℐ1∨ℐ2A⊆Buℐ1∨ℐ2ACorollary 6.
LetU,R,ξj be a j–NS, ℐ1∨ℐ2 an ideal on U, and A⊆U. Then, ∀j∈r,l,r,l,i,u,i,u:(1)
σuℐ1∨ℐ2A≤σrℐ1∨ℐ2A≤σiℐ1∨ℐ2A(2)
σuℐ1∨ℐ2A≤σlℐ1∨ℐ2A≤σiℐ1∨ℐ2A(3)
σuℐ1∨ℐ2A≤σrℐ1∨ℐ2A≤σiℐ1∨ℐ2A(4)
σuℐ1∨ℐ2A≤σlℐ1∨ℐ2A≤σiℐ1∨ℐ2ARemark 8.
LetU,R,ξj be a j–NS and ℐ1∨ℐ2 an ideal on U. Then, the following should be noted that(1)
σrℐ1∨ℐ2A and σrℐ1∨ℐ2A are not necessarily to be comparable(2)
σlℐ1∨ℐ2A and σlℐ1∨ℐ2A are not necessarily to be comparable(3)
σiℐ1∨ℐ2A and σiℐ1∨ℐ2A are not necessarily to be comparable(4)
σuℐ1∨ℐ2A and σuℐ1∨ℐ2A are not necessarily to be comparableRemark 9.
Table1 shows the following results:(1)
The comparison among theℐ1∨ℐ2-j-lower, ℐ1∨ℐ2-j-upper approximations, ℐ1∨ℐ2-j-boundary regions, and ℐ1∨ℐ2-j-accuracy in Definition 14 by using Example 1, for j∈r,l,i,u.(2)
There are different methods to approximate the sets by usingτjℐ1∨ℐ2,∀j∈r,l,i,u in constructing the approximations of sets. The best one of these methods is given by using τiℐ1∨ℐ2 in constructing the approximations of sets since the boundary regions, in this case, are decreased (or canceled) by increasing the lower approximations and decreasing the upper approximations. Moreover, the ℐ1∨ℐ2-i-accuracy is more accurate than the other types since σuℐ1∨ℐ2A≤σrℐ1∨ℐ2A≤σiℐ1∨ℐ2A and σuℐ1∨ℐ2A≤σlℐ1∨ℐ2A≤σiℐ1∨ℐ2A.
It should be noted that the same as in Table1 we can add an example to show the comparison among the ℐ1∨ℐ2-j-lower, ℐ1∨ℐ2-j-upper approximations, ℐ1∨ℐ2-j-boundary regions, and ℐ1∨ℐ2-j-accuracy in Definition 14, for j∈r,l,i,u, and to illustrate that there are different methods to approximate the sets by using τjℐ1∨ℐ2,∀j∈r,l,i,u in constructing the approximations of sets. The best one of these methods is given by using τiℐ1∨ℐ2 in constructing the approximations of sets since the boundary regions in this case are decreased (or canceled) by increasing the lower approximations and decreasing the upper approximations. Moreover, the ℐ1∨ℐ2-i-accuracy is more accurate than the other types since σuℐ1∨ℐ2A≤σrℐ1∨ℐ2A≤σiℐ1∨ℐ2A and σuℐ1∨ℐ2A≤σlℐ1∨ℐ2A≤σiℐ1∨ℐ2A.Remark 10.
It should be noted that the relationships among theℐ1,ℐ2-j-lower, ℐ1,ℐ2-j-upper approximations, ℐ1,ℐ2-j-boundary regions, and ℐ1,ℐ2-j-accuracy are the same as in Propositions 6 and 7 and Corollaries 5 and 6.


## 5. Application

Example 2.
LetU=x1,x2,x3,x4,x5 be five amino acids (for short, AAs). The AAs are described in terms of seven attributes: a1 = PIE, a2 = PIF (two measures of the side chain lipophilicity), a3 = DGR = ΔG of transfer from the protein interior to water, a4 = SAC = surface area, a5 = MR = molecular refractivity, a6 = LAM = the side chain polarity, and a7 = Vol = molecular volume [36,37]. Table 3 shows all quantitative attributes of five AAs.Table 3
Quantitative attributes of five amino acids.
a1a2a3a4a5a6a7x1−0.11−0.220.293353.458−1.19127.5x2−0.51−0.640.76311.63.243−1.43120.5x3000224.91.6620.0365x40.150.13−0.25337.23.856−1.06140.6x51.21.8−2.1322.63.350.04131.7We consider the following seven relations onU defined as follows: Ri=xi,xj:xiak−xjak<σk/2,i,j=1,2,…,5,k=1,2,…,7 where σk represents the standard deviation of the quantitative attributes ak,k=1,2,…,7. The right neighborhoods for all elements of U=x1,x2,x3,x4,x5 with respect to the relations Rk,k=1,2,…,7 are shown in Table 4.Table 4
Right neighborhood of the seven relations.
xiR1uiR2uiR3uiR4uiR5uiR6uiR7x1x1,x3,x4,x5Ux1,x2,x3,x4x1,x4,x5x1,x2,x4,x5Ux1,x2,x4,x5x2UUx1,x2x1,x2,x4,x5x1,x2,x4,x5Ux1,x2,x4,x5x3x1,x3,x4,x5x1,x3,x4,x5x1,x2,x3,x4UUx3,x5Ux4x1,x3,x4,x5x1,x3,x4,x5x1,x2,x3,x4x1,x4,x5x1,x4x1,x3,x4,x5x1,x4,x5x5x5x5Ux1,x2,x4,x5x1,x2,x4,x5x3,x5x1,x2,x4,x5Therefore, we find the intersection of all right neighborhoods of all elementk=1,2,…,7 as follows: x1R=∩k=17x1Rk=x1,x4,x2R=∩k=17x2Rk=x1,x2,x3R=∩k=17x3Rk=x3,x4R=∩k=17x4Rk=x1,x4,x5R=∩k=17x5Rk=x5. Then, R=Δ∪x1,x4,x2,x1,x4,x1, where Δ is the identity relation and equal to x1,x1,x2,x2,x3,x3,x4,x4,x5,x5. Let ℐ1=ϕ,x4,ℐ2=ϕ,x1, and ℐ1∨ℐ2=ϕ,x1,x4,x1,x4. Then, we calculate the lower, upper approximations, boundary region, and accuracy by using Abd El-Monsef et al.’s Definition 6 [35] and Hosny’s Definition 9 [28] and the current method in Definitions 12 and 14 as shown in Table 5. Hence, for any concept A⊆U (collection of amino acid), this concept is determined by lower and upper approximations which define its boundary. The accuracy increases by the decrease of the boundary region. Clearly, the accuracy measure by using the current approximations in Definitions 12 and 14 is greater than the accuracy measure by using Abd El-Monsef et al.’s Definition 6 [35] and Hosny’s Definition 9 [28].Table 5
Comparison between the boundary and accuracy by using the current approximations in Definitions12 and 14 and the previous ones in Definitions 6 [35] and 9 [28] at j=r and k=1.
AThe previous one in Definition6 [35]The previous one in Definition9 [28]The current method in Definition12The current method in Definition14R¯rAR¯rABrAσrAR¯rℐ1AR¯rℐ1ABrℐ1Aσrℐ1AR¯rℐ1,ℐ2AR¯rℐ1,ℐ2ABrℐ1,ℐ2Aσrℐ1,ℐ2AR¯rℐ1∨ℐ2AR¯rℐ1∨ℐ2ABrℐ1∨ℐ2Aσrℐ1∨ℐ2Ax1ϕx1,x2,x4x1,x2,x40x1x1,x2,x4x2,x41/3x1x1ϕ1x1x1ϕ1x2ϕx2x20ϕx2x20x2x2ϕ1x2x2ϕ1x3x3x3ϕ1x3x3ϕ1x3x3ϕ1x3x3ϕ1x4ϕx1,x2,x4x1,x2,x40ϕx4x40x4x4ϕ1x4x4ϕ1x5x5x2,x5x21/2x5x5ϕ1x5x5ϕ1x5x5ϕ1x1,x2ϕx1,x2,x4x1,x2,x40x1,x2x1,x2,x4x42/3x1,x2x1,x2ϕ1x1,x2x1,x2ϕ1x1,x3x3x1,x2,x3,x4x1,x2,x41/3x1,x3x1,x2,x3,x4x2,x41/2x1,x3x1,x3ϕ1x1,x3x1,x3ϕ1x1,x4ϕx1,x2,x4x1,x2,x40x1,x4x1,x2,x4x22/3x1,x4x1,x4ϕ1x1,x4x1,x4ϕ1x1,x5x5x1,x2,x4,x5x1,x2,x41/4x1,x5x1,x2,x4,x5x2,x41/2x1,x5x1,x5ϕ1x1,x5x1,x5ϕ1x2,x3x3x2,x3x21/2x3x2,x3x21/2x2,x3x2,x3ϕ1x2,x3x2,x3ϕ1x2,x4ϕx1,x2,x4x1,x2,x40ϕx2,x4x2,x40x2,x4x2,x4ϕ1x2,x4x2,x4ϕ1x2,x5x5x2,x5x21/2x5x2,x5x21/2x2,x5x2,x5ϕ1x2,x5x2,x5ϕ1x3,x4x3x1,x2,x4,x5x1,x2,x41/4x3x3,x4x41/2x3,x4x3,x4ϕ1x3,x4x3,x4ϕ1x3,x5x3,x5x3,x5ϕ1/4x3,x5x3,x5ϕ1/2x3,x5x3,x5ϕ1x3,x5x3,x5ϕ1x4,x5x5x1,x2,x4,x5x1,x2,x41/4x5x4,x5x41/2x4,x5x4,x5ϕ1x4,x5x4,x5ϕ1x1,x2,x3x3x1,x2,x3,u4x1,x2,x41/4x1,x2,x3x1,x2,x3,x4x43/4x1,x2,x3x1,x2,x3ϕ1x1,x2,x3x1,x2,x3ϕ1x1,x2,x4x1,x2,x4x1,x2,x4ϕ1/4x1,x2,x4x1,x2,x4ϕ1x1,x2,x4x1,x2,x4ϕ1x1,x2,x4x1,x2,x4ϕ1x1,x2,x5x5x1,x2,x4,x5x1,x2,x41/4x1,x2,x5x1,x2,x4,x5x43/4x1,x2,x5x1,x2,x5ϕ1x1,x2,x5x1,x2,x5ϕ1x1,x3,x4x1,x3,x4x1,x2,x3,x4x23/4x1,x3,x4x1,x2,x3,x4x23/4x1,x3,x4x1,x3,x4ϕ1x1,x3,x4x1,x3,x4ϕ1x1,x3,x5x3,x5Ux1,x2,x42/5x1,x3,x5Ux2,x43/5x1,x3,x5x1,x3,x5ϕ1x1,x3,x5x1,x3,x5ϕ1x1,x4,x5x1,x4,x5x1,x2,x4,x5x23/4x1,x4,x5x1,x2,x4,x5x23/4x1,x4,x5x1,x4,x5ϕ1x1,x4,x5x1,x4,x5ϕ1x2,x3,x4x3x1,x2,x3,x4x2,x3,x41/4u3u2,u3,u4u2,u41/3u2,u3,u4u2,u3,u4ϕ1u2,u3,u4u2,u3,u4ϕ1x2,x3,x5x3,x5x2,x3,x5x22/3x3,x5x2,x3,x5x22/3x2,x3,x5x2,x3,x5ϕ1x2,x3,x5x2,x3,x5ϕ1x2,x4,x5x5x1,x2,x4,x5x1,x2,x41/4x5x2,x4,x5x2,x41/3x2,x4,x5x2,x4,x5ϕ1x2,x4,x5x2,x4,x5ϕ1x3,x4,x5x3,x5Ux1,x2,x42/5x3,x5x3,x4,x5x42/3x3,x4,x5x3,x4,x5ϕ1x3,x4,x5x3,x4,x5ϕ1x1,x2,x3,x4x1,x2,x3,x4x1,x2,x3,x4ϕ1x1,x2,x3,x4x1,x2,x3,x4ϕ1x1,x2,x3,x4x1,x2,x3,x4ϕ1x1,x2,x3,x4x1,x2,x3,x4ϕ1x1,x2,x3,x5x1,x3,x5Ux2,x43/5x1,x2,x3,x5Ux44/5x1,x2,x3,x5x1,x2,x3,x5ϕ1x1,x2,x3,x5x1,x2,x3,x5ϕ1x1,x2,x4,x5x1,x2,x4,x5Ux34/5x1,x2,x4,x5x1,x2,x4,x5ϕ1x1,x2,x4,x5x1,x2,x4,x5ϕ1x1,x2,x4,x5x1,x2,x4,x5ϕ1x1,x3,x4,x5x1,x3,x4,x5Ux24/5x1,x3,x4,x5Ux24/5x1,x3,x4,x5x1,x3,x4,x5ϕ1x1,x3,x4,x5x1,x3,x4,x5ϕ1x2,x3,x4,x5x3,x5Ux1,x2,x42/5x3,x5x2,x3,x4,x5x2,x41/2x2,x3,x4,x5x2,x3,x4,x5ϕ1x2,x3,x4,x5x2,x3,x4,x5ϕ1UUUϕ1UUϕ1UUϕ1UUϕ1


## 6. Conclusions

The rough theory is a new mathematical tool to deal with vagueness and imperfect knowledge by using the concept of the lower and upper approximations. If the lower and upper approximations of the set are equal to each other, then it is called a crisp (exact) set; otherwise, it is known as a rough (inexact) set. Therefore, the boundary region is defined as the difference between the upper and lower approximations, and then the accuracy of the set or ambiguous depending on the boundary region is empty or not respectively. A nonempty boundary region of a set means that our knowledge about the set is not sufficient to define the set precisely. The main aim of the rough set is to reduce the boundary region by increasing the lower approximation and decrease the upper approximation. An interesting and natural research topic in the rough set theory is to study the rough set theory via topology. The topology induced by binary relations on the universes and bi-ideals was used to generalize the basic rough set concepts. In this paper, different new types of rough set based on bi-ideals were defined to reduce the boundary region and increase the accuracy measure which is the main aim of rough set theory. The concepts of lower and upper approximations based on bi-ideals were presented for these types. Several properties and examples were provided. Additionally, some important properties and results of these approximations were instituted. The relationships between the present approximations and the previous approximations were established and shown to be more general. We conclude that the suggested topological operations and structures open the way for more topological applications in rough context. We can say that the improvements in abstract topology results help in some way in the modifications of rough sets theory and consequently in its real-life applications. It should be noted that any one can extend the current method similarly by usingn-ideals. In the real life, this comparison is represented if we need to make a decision about a manuscript and send it to two reviewers ℐ1,ℐ2. Then, we have two cases:(1)
Take two reports separately from the referees and use one of the mathematical methods to find the final decision(2)
Take a combined report from the referees togetherIt is clear that the second case is better than the first one such as the approximations of the present study.



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