Approximation spaces inspired by subset rough neighborhoods with applications

: In this manuscript, we ﬁ rst generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough - set models. Then, we present other types of generalized rough - set models directly de ﬁ ned by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the ﬁ rst one in cases of u and u ⟨ ⟩ , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approx - imation compared to the existing approaches in published literature. Algorithms and a ﬂ ow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to con ﬁ rm the e ﬃ ciency of our approaches to maximize the value of accuracy and shrink the boundary regions.


Introduction
Rough-set theory is a novel mathematical approach originated by Pawlak [1] in the 1980s to manage inexplicit and uncertain data that cannot be addressed by the classical set theory.The key idea in this approach is the approximation space (AS) which comprises an equivalence relation on a nonempty set of objects.By Pawlak's approach, each subset of data can be approximated using approximation operators called lower approximation and upper approximation, which are defined by the equivalence classes induced by .These operators categorize the knowledge obtained from the data into three main regions: positive, negative, and boundary.
In many real-life issues that humans deal with in computer networks, economics, medical sciences, engineering, etc., the condition of an equivalence relation does not appear as a description for the relationship between the objects, which abolishes the ability of Pawlak's rough-set theory to deal with these problems [2].To overcome this obstacle various frames of rough set theory defined with respect to nonequivalence relations, known as generalized rough-set theory or generalized AS, have been proposed.
The first generalized rough-set model constructed by a non-equivalence relation was introduced by Yao [3] in 1996.He defined the concepts of "right neighborhood N r " and "left neighborhood N l " of each object under arbitrary relation as alternatives to the equivalence class.That is, the granules or blocks that are used to approximate the knowledge obtained from the subset of data are these types of neighborhoods.Then, researchers have established other kinds of generalized ASs under specific relations like tolerance [2], similarity [4,5], quasiorder [6,7] and dominance [8,9].It has been introduced that many generalized ASs are produced by specific kinds of neighborhood systems; for example, Dai et al. [10] scrutinized some models of ASs using the maximal right neighborhoods defined over a similarity relation.Al-shami [11] completed studying the other kinds of maximal neighborhoods under any arbitrary relation and showed how they applied to classify patients suspected of infection with COVID-19.
To improve the approximation operators by adding objects to the lower approximation and/or removing objects from the upper approximation, the concepts of core neighborhoods and remote neighborhoods were presented by Mareay [12] and Sun et al. [13], respectively.Also, Abu-Donia [14] adopted a new line of rough-set models depending on a finite family of arbitrary relations instead of one relation.Recently, Al-shami with his co-authors have displayed novel sorts of neighborhood systems and their generalized rough paradigms inspired by some relationships between N ρ -neighborhoods, such as C ρ -neighborhoods [15], S ρ -neighborhoods [16], and E ρ -neighborhoods [17].
Topology is another interesting orientation for studying rough-sets.The possibility of replacing roughset concepts with their topological counterparts follows from the similar behaviors of topological and rough-set concepts.Investigation of this link was started by Skowron [18] and Wiweger [19].This domain attracted many scholars and researchers to initiate rough-set notions via their topological counterparts; for instance, Lashin et al. [20] suggested a family N ρ -neighborhood of each element as a subbase for topology, and then they coped with the notions of rough-set theory as topological concepts.Salama [21] debated how the missing attribute value problem is solved topologically.Al-shami [22,23] benefited from somewhere dense and somewhat open subsets of topological spaces to present various types of approximation operators and accuracy measures.Al-shami and Alshammari [24] successfully applied the structure of supra topology, one of the generalizations of topology, to study generalized rough ASs.To complete this line of research and enhance the role of generalizations of topology to describe the main concepts of rough sets, Al-shami and Mhemdi [25] investigated the rough approximation operators via the frame of infra topology, and the authors of [26,27] discussed these operators via minimal structures.Many ideas and relationships that associated rough-set models with topological counterparts have been elucidated and reveled in [28][29][30][31][32][33].
In 2013, Kandi et al. [34] provided a novel method to construct ASs depending on the structure of an ideal.They aimed to improve approximation operators and increase accuracy measures.Then, Hosny [35] introduced new rough-set models induced from topological and ideal structures.Recently, some types of neighborhoods with ideal structures have been applied to obtain rid of uncertainty via information systems in [36][37][38][39][40].
The major motivations for writing this article are, first, to dispense an equivalence relation that limits the applications of Pawlak rough-set theory.Second, to keep the greatest number of properties of Pawlak approximation operators that are missing in some existing approaches generated by topological approaches or otherwise.Third, to maximize the accuracy measures and minimize the boundary regions of subsets compared to the previous approaches introduced in [11,40] under arbitrary relation and those introduced in [17,35,41] under similarity relation.
The rest of this article is designed as follows.In Section 2, we review the main concepts of rough sets and topology required to understand this work and shed light on the inducements that led to these contributions.In Section 3, we provide a method to build some topologies by using subset neighborhoods and ideals with respect to any arbitrary relation.Then, we establish new generalized rough-set models by making use of these topologies and discuss their fundamental characterizations.In Section 4, we construct the counterparts of the previous generalized rough set and elucidate their advantages to develop the approximation operators.In addition, we provide an algorithm illustrating how to determine exact sets.We analyze the information system of dengue fever disease in Section 5 to demonstrate the effectiveness and robustness of the followed approach to maximize accuracy values and shrink boundary regions.
Finally, we summarize the main contributions and give some thoughts that can be applied to expand the scope of this manuscript in Section 6.

Preliminaries
To make the exposition self-contained, we mention in this section some basic concepts and results of rough-set theory and topology used in the sequel.In this work, the order pair , ( ) The next proposition outlines the essential properties of these approximation operators, which is the key point of rough-set theory.
Proposition 2.2.[1] Let V and W be subsets of a Pawlak AS , ( ) .Then, we have next properties.
Every subset of data is divided into three regions using approximation operators, aiming to discover the knowledge obtained from a subset and its structure.

Definition 2.3. [1]
We associate every subset V of a Pawlak AS , ( ) with three regions defined as follows: The measure (or completeness degree) of knowledge obtained from a nonempty subset V is given as follows: To expand the applications of rough-set theory, Yao [3,42] replaced the equivalence relation with arbitrary relation.In this situation, we need a counterpart for the equivalence classes as a granule for computing.So, it was defined as "right and left neighborhoods," which play a role in equivalence classes in Pawlak AS. ) be an AS.Then, the right neighborhood N r and left neighborhood N l of ν ∈ are, respectively, given as follows: N ν μ ν μ N ν μ μ ν : , and : , .
The approximation operators were formulated in view of right and left neighborhoods as follows.
Definition 2.5.[3,42] It was introduced the Nρ-lower and Nρ-upper approximations of a subset V of an AS , ( ) for ρ r l , { } ∈ as follows: Subsequently, researchers and scholars have investigated various forms of generalized rough sets inspired by new neighborhood systems, aiming to improve the approximations and increase the accuracy measures of rough subsets.In what follows, we list some of them.Definition 2.6.[4,43,44] The ρ-neighborhoods of each ν ∈ , denoted by N ν ρ ( ), induced from an AS , ( ) are given for ρ r l i u i u , , , , , Following a similar technique of Definition 2.5, the above neighborhoods were employed to present new sorts of approximation operators.
But using the formula of Pawlak accuracy measures leads sometimes to obtaining values greater than one, which is illogical.To remove this failure, the definition of accuracy measures was adjusted as follows.
Definition 2.7.[3,4,[42][43][44] The accuracy measures of a nonempty subset V in an AS , ( ) is given as follows: Two of the celebrated types of rough neighborhoods are E ρ -neighborhoods and S ρ -neighborhoods.They were defined as follows.
[16] The S ρ -neighborhoods of each ν ∈ , denoted by S ν ρ ( ), induced from an AS , ( ) are given for ρ r l r l i u i u , , , , , , , Definition 2.11.[16,17] The lower and upper approximations and accuracy measure of a subset V of an AS , ( ) for each ρ are defined with respect to E ρ -neighborhoods and S ρ -neighborhoods as follows: To minimize the vagueness of the data by decreasing the upper approximation and increasing the lower approximation, the approximation operators were constructed from an ideal structure.We first mention the definition of ideal and then present how this idea was exploited to produce new operators of approximation.
Definition 2.12.We call a non-empty family of 2 an ideal on if it is closed under subset and finite union.That is, it satisfies the following axioms: Henceforth, we call the triplet , , ( ) an ideal approximation space (IAS).
[34] The lower and upper approximations and accuracy measure of a subset V of an IAS , , ( ) for each ρ are defined with respect to N ρ -neighborhoods as follows: , where .
14. [41] The lower and upper approximations and accuracy measure of a subset V of an IAS , , ( ) for each ρ are defined with respect to E ρ -neighborhoods as follows: Rough approximation spaces  5 , where .
A subfamily Ω of P( ) is called a topology on if ϕ, Ω ∈ , and it is closed under arbitrary union and finite intersection.We call an order pair , Ω ( ) a topological space.We call a member of Ω an open set and call the complement of an open set a closed set.
For any subset V of , the interior points of V , denoted by V int( ), is the union of all open sets that are contained in V , and the closure points of V , denoted by V cl( ), is the intersection of all closed sets con- taining V .
The rough-set paradigms have been studied topologically in several published literature.The followed methods to link neighborhoods systems and topological structures are proved in the following results.
Theorem 2.16.Let , ( ) be an AS.Then, each one of the following families is a topology on for each ρ: The aforementioned topological spaces have been employed to construct novel types of ASs.
Definition 2.17.[17,40,43] Let , ( ) be an AS.Then, some types of lower and upper approximations and accuracy measures of a subset V ⊆ induced from topological spaces Ω Nj and Ω Nj are, respectively, defined as follows: To improve the approximation operators and increase the accuracy measure of a set, the topological structures given in Theorem 2.16 were enlarged by inserting ideals as illustrated in the next theorem.
) be an IAS.Then, each one of the following families is a topology on for each ρ: ) be an IAS.Then, some types of lower and upper approximations and accuracy measures of a subset V ⊆ induced from topological spaces Ω Nj and Ω Ej are, respectively, defined as follows: 3 ASs generated by subset topologies and ideal In this section, we provide a new method to propose a topological approach to construct ASs inspired by the ideas of subset neighborhoods and ideals.First, we illustrate the relationships between them and explore their main characterizations.Then, we confirm the good performance of the proposed approach in terms of improving the accuracy measures and approximation operators compared to some previous methods introduced in [16] for ρ r l i r l i , , , , { } ∈ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ and [40] for each ρ under any arbitrary relation.Furthermore, the current approach is more accurate than that of Hosny's [35] under a similarity relation and Yildirim's [41] under a reflexive relation for each ρ.Finally, we give an algorithm and flow chart to illustrate how the exact and rough sets are determined.
) be an IAS.Then, the family According to the hereditary property of , we obtain S ν X Hence, the proof is complete.□ The following result elaborates on the relationships between these topologies.
Theorem 3.2.The next results hold true. ( If is a quasiorder (reflexive and transitive), then Ω Ω Proof.The proofs of items 1-4 follows from the below relationships and the hereditary property of the ideal.

S ν S ν S ν S ν S ν S ν S ν S ν S S ν S ν S
; ; and .
hold true under a symmetric relation and the equality hold true under an equivalence relation, we obtain the proofs of the results (5) and (6).
The proof of result (7) follows from the fact that N ν N ν ρ ρ ( ) ( ) = ⟨ ⟩ for each element ν under a quasiorder relation. □ Now, we benefit from the topological structures constructed in Theorem 3.1 to initiate novel types of ASs. ) an ideal subset topological approximation space (ISTAS), where Ω Sρ is the topological space obtained from Theorem 3.1.A subset V of an ISTAS , , Ω Sρ ( ) is said to be an S ρ -open set if X Ω Sρ ∈ , and the complement of an S ρ -open set is said to be an S ρ -closed set.The family ϒ Sρ of all S ρ -closed sets is given as follows: ) be an ISTAS.The S ρ -interior and S ρ -closure of a subset V of are, respectively, defined as follows: The main concepts of ASs are defined with respect to an ISTAS , , Ω Sρ ( ) as follows.
Definition 3.5.Let , , Ω Sρ ( ) be an ISTAS.The lower approximation S ρ , upper approximation S ρ , boundary region B Sρ , positive region O Sρ + , negative region O Sρ − , and accuracy measure λ Sρ of a subset V are, respectively, given as follows: - The following theorem states the properties of the lower approximation S ρ and upper approxima- tion S ρ .
Theorem 3.6.Consider , , Ω Sρ ( ) as an ISTAS and let V and W be subsets of .Then, ( ( Proof.The proof follows from the properties of interior and closure topological operators.□ Proposition 3.7.Let , , Ω Sρ ( ) be an ISTAS and V ⊆ .Then, Proof.We suffice by proving (i), and the other cases can be proved following a similar technique.Let . Then, it follows from Theorem 3.2 that ν V int Sr ( ) ∈ . This means that . In a similar way, we obtain . Hence, the proof is complete.□ Corollary 3.8.Let V be a nonempty subset of an ISTAS , , Ω Sρ ( ) .Then, Proof.We suffice by proving (i), and the other cases can be proved following a similar technique.To do this, note that . This automatically leads to the next equality In addition, note that , which automatically leads to the following equality: By (1) and (2), we obtain which ends the proof.□ In the following proposition and example, we show that the the current ISTASs are more efficient at removing vagueness than their counterparts given in [40].
for each subset V of .This automatically leads to the following equalities: It comes from (3) and (4) that ) be an AS, where { } = be a universe set, and δ δ δ δ , , , , )}be a binary relation on .We compute the systems of N ρ -neighborhoods and S ρ -neighborhoods in Tables 1 and 2, respectively.First, we generate eight topologies from the system of S ρ -neighborhoods.
S u In Tables 3-5, we compute the approximation operators S ρ and S ρ and accuracy measures λ Sρ of each subset of produced by the approach given in Definition 3.5.
Table 1: Table 3: S ρ -approximations and generate eight topologies from the system of S ρ -neighborhoods and ideal .
In Tables 6-8, we compute the approximation operators S ρ and S ρ and accuracy measures λ Sρ of each subset of produced by the current approach given Definition 3.5.
It can be seen from Tables 3-8 that the current approach maximizes the lower approximation and minimizes the upper approximation; hence, heightening the value of accuracy.This gives an advantage for our approach in terms of improving the approximation operators and increasing the accuracy measure of a subset compared to the approach studied in [40] for all cases of ρ under any binary relation.The cells given in bold in Tables 6-8 validate this fact.
Remark 3.11.It was proved in Proposition 6 of [40] that the topological approaches and their counterparts given in [16] are identical for ρ r l i r l i , , , , { } ∈ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ .So, according to the previous results and example presented herein, we infer that the current approach is also better than the approach given in [16] for ρ r l i r l i , , , , Remark 3.12.The current approach and the approach introduced in [37] are independent of each other.This matter can be confirmed by the computations given in Tables 6-8, and computing their counterparts induced by the system of containment neighborhoods.
In the next two results, we point out that the current approach is better than the approach discussed in [41] under a reflexive relation.Following similar arguments, one can prove the next two results, which show the advantages of the current approach compared to the approaches displayed in [35,41] to improve approximation operators and increase the value of accuracy.Proposition 3.15.Let V be a subset of an IAS , , ( ) such that is similarity.Then . Hence, the proof is complete.□  ) ) is S ρ -exact iff V ϕ ρ ( ) = .which means that V is S ρ -exact.□ In the end of this section, we provide Algorithm 1 and Figure 1 to illustrate how it can be determined whether a subset of an IAS , , ( ) is S ρ -exact or S ρ -rough.
Algorithm 1: The algorithm of determining S ρ -exact and S ρ -rough sets in an IAS , , ( ) .
Output: Classification a set in an IAS , , ( ) into two categories: S ρ -exact or S ρ -rough.4 ASs generated by subset neighborhoods and ideal In this section, we give another method to produce ASs from subset neighborhoods and ideals in a direct way.We reveal the relationships between them and discuss their essential features.Also, we mention the characterizations of Pawlak approximation operators that are missing via the current approach.Compared to the approach given in the previous section, we demonstrate that the current approach improves the comparisons between the current approaches and previous ones introduced in [16,17,35,40,41] have been conducted under different types of binary relations aiming to show the advantages of the current approaches to maximize the accuracy measure of a subset by increasing lower approximation and decreasing upper approximation.
To confirm that there is a need for investigation considering the various sorts of neighborhoods systems so that these findings may contribute to remove uncertainty of real data, we examine and analyze the performance of the current methods and some foregoing ones via the information system of dengue fever.The obtained results concluded that the approaches proposed herein were more general and accurate.
In upcoming studies, we will adopt another type of neighborhoods to obtain rid of vagueness in data.Also, we will research the current models depending on a finite family of arbitrary relations instead of one relation.Moreover, we discuss the presented approaches in the content of soft rough set and fuzzy rough set.

Figure 1 :
Figure 1: Flow chart of determining S ρ -exact and S ρ -rough sets.

1each ρ do 8 |
Specify a relation over the universal set ; Build a topology Ω ρ using Theorem 3

Table 5 :
λ Sρ -accuracy measures This means that W is also an open subset in Ω Sρ .Thus, ν S V

Table 8 :
λ Sρ -accuracy measures 3.16.Let V be a subset of an IAS