Abstract
One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article to generate some kinds of topologies from ideals and -neighborhoods which are induced from any binary relation. We define new types of approximations and accuracy measures from these topologies and then compare them with their counterparts induced directly from -neighborhoods and ideals. Also, we show that the approximations and accuracy measures given, herein, are better than those introduced in some previous studies under any arbitrary relation.
1. Introduction
Rough set theory [1, 2]is one of the followed methods to handle the vagueness (uncertainty) of the information systems data and imperfect knowledge. In this theory, each subset is associated with two crisp sets (called lower and upper approximations) generated from an equivalence relation. To extend the applications of rough set theory, many authors have replaced an equivalence relation by different kinds of relations.
The interaction between topological and rough set theory is due to Skowron [3] and Wiweger [4] who first discussed the role of topological aspects in rough sets. Then, a combination of rough set theory and topological theory became the main goal of many studies [5–11]. This interaction also included some generalizations of topology such as minimal structure [12]. We refer the reader to [13] to see the main contributions which investigated the relationships between topology and rough set theory.
Ideals in a topological space have been taken into account by Kuratowski [14] and defined as a nonempty collection of subsets of a universe which is closed under finite union and subsets. Kandil et al. [15] applied the notion of ideals with -neighborhoods to generalize Pawlak’s approximations. They showed that their results decrease the boundary region in comparison with Pawlak’s method [2], Allam’s method [16], and Yao’s method [17].
Recently, some new types of neighborhood systems have been introduced and studied. Among them, -neighborhood systems [18] and -neighborhood systems [19] are studied. In [18], Al-Shami et al. exploited -neighborhoods to establish new rough and topological approximations. They compared between them and showed that the accuracy measure obtained from rough approximations is better than their counterparts obtained from topological approximations.
Through this study, we first construct new topological spaces using the ideas of -neighborhood and ideals to minimize the boundary region and maximize the accuracy measure of a set compared with the approaches introduced in [18]. Second, we establish new rough approximations that improved the topological approximations.
2. Preliminaries
In this section, we recollect several substantial features and outcomes of rough set theory, especially those regarding to some sorts of neighborhood systems.
Definition 1. (1)[1] A binary relation on (i.e., ) is said to be(i)Equivalence if it is reflexive (i.e., for each ), symmetric (i.e., if ), and transitive (i.e., , whenever and )(ii)Tolerance if it is reflexive and symmetric(iii)Preorder (quasiorder or dominance) if it is reflexive and transitive(iv)Partial order if it is a antisymmetric (i.e., , whenever and ) preorder(v)Diagonal if (vi)Serial if every , there exists , such that (2)[2] For an equivalence relation on and a subset , the two related sets and are called lower approximation and upper approximation of , respectively.
Definition 2 (see [16, 17, 20]). Let be a binary relation on . The -neighborhoods of (, in short) are defined for each as follows:(1)-neighborhood: (2)-neighborhood: (3)-neighborhood: (4)-neighborhood: (5)-neighborhood: provided that there exists containing . Otherwise, .(6)-neighborhood: provided that there exists containing . Otherwise, .(7)-neighborhood: (8)-neighborhood: Henceforth, for all the following results, we will deal with all the values of , , unless otherwise noted.
Definition 3 (see [20]). Let be a binary relation on and : be a mapping which assigns for each in its -neighborhood in . Then, is called a -neighborhood space (briefly, ).
Proposition 1 (see [18]). Let be and . Then,(1)If is a reflexive relation, then , for all (2)If is a transitive relation, then , for all (3)If is a symmetric relation, then
Theorem 1 (see [20]). Let be . Then, for each , the collection is a topology on .
Definition 4 (see [18]). Let be . Then, a set is called -open set if , and its complement is called -closed set. The family of all -closed sets of a -neighborhood space is defined by , where is the complement of .
Definition 5 (see [18]). Let be an arbitrary binary relation on . The -neighborhood of (, in short) is defined for each as follows:(1)(2)(3)(4)(5)(6)(7)(8)
Theorem 2 (see [18]). Let be an arbitrary binary relation on and . Then, -neighborhoods have the following properties:(1) iff (2)If is reflexive, then and , for all (3)If is symmetric, then and (4)If is transitive, then , for all (5)If is symmetric and transitive, then and (if ), for each (6)If is preorder, then , for all (7)If is an equivalence relation, then for each , all are identical, , and iff
In [18], Al-Shami et al. formulated the concepts of -lower and -upper approximations and -accuracy measure of a subset in terms of -neighborhoods as follows:
Definition 6 (see [18]). Let be a subset of an . Then, the -lower and -upper approximations and -accuracy measure of a subset are(1)(2)(3), where
Theorem 3 (see [18]). Let be and . Then, for each , the following properties hold:(i) and (ii) and (iii)If , then and (iv) and (v) and (vi) and (vii)Generally, and
Also, Al-Shami et al. [18] employed -neighborhoods to generate various topologies and studied their basic characteristics. Furthermore, they introduced the concepts of -lower and -upper approximations and -accuracy measure of a subset induced from these topologies.
Theorem 4 (see [18]). Let be . For each , the collection is a topology on .
Definition 7 (see [18]). Let be . A set is called -open set if , and its complement is called -closed set. The family of all -closed sets of a -neighborhood space is defined by .
Theorem 5 (see [18]). Let be an arbitrary binary relation on and . If is an equivalence relation, then .
Definition 8 (see [18]). Let be a topology generated by -neighborhoods. If and , then the -lower and -upper approximations and -accuracy of are defined, respectively, as(1), where represents the interior points of w.r.t. (2), where represents closure points of w.r.t. (3), where
Theorem 6 (see [21]). Let be and be an ideal on . If , then the collection is a topology on .
3. Sorts of Approximations Based on -Neighbourhoods and Ideals
Al-Shami et al. [18] constructed approximations relying on various topologies that are induced from the four types of -neighbourhoods. In this portion, we shall generalize these topologies by using ideals and deduce new rough approximations based on -neighbourhoods and ideals. We explain the relationships between these approximations and provide illustrative examples.
3.1. Various Topologies Generated from -Neighbourhoods via Ideals
In this part, we employ -neighborhoods and ideal to generate various topologies that are finer than the previous one generated by -neighborhoods due to [18] for any relation.
First, we are going to offer a method of generating some topologies by using -neighborhoods and ideal .
Theorem 7. Let be . For each , the collection is a topology on .
Proof. Let , , and , then there exists s.t. . Hence, . Since , then , i.e., .
Let and . Then, and . According to properties of , . Hence, . It follows that .
Easily, , . Consequently, is a topology on .
Definition 9. Let be and be an ideal on . A set is called -open set if , and the complement of -open set is called -closed set. The family of all -closed sets is defined by .
Theorem 8. Let be and be an ideal on . Then,(1)(2)If is a reflexive relation and , then (3)If is a symmetric, then and (4)If is a transitive relation and , then (5)If is a preorder relation and , then (6)If is an equivalence relation, then for each , all are identical, .
Proof. In view of Theorem 2, then the proof is obvious.
The next proposition shows that the relation between the topologies and -neighborhoods is reversible for each .
Proposition 2. Let be and be an ideal on . Then, the following results hold.(1)(2)
Proof. Since and , where or , then the proof is obvious.
Example 1. Let and be a binary relation on . In Tables 1 and 2 , we calculate -neighborhoods and -neighborhoods for each element of .
Lemma 1. Let and . Then, for any binary relation on , .
Proof. Obvious.
Lemma 2. Let be two ideals on . If , then for each .
Proof. Straightforward.
The following example shows that the inclusion in Proposition 2 and Lemma 2 cannot be replaced by the equality relation.
Example 2. Continued from Example 1.If , thenIf , then
3.2. Generalized Rough Approximations Based on Various Topologies Generated from -Neighbourhoods via Ideals
Herein, we will construct some kinds of rough approximations using the topologies generated from -neighborhoods and ideals and give some properties of them.
Definition 10. Let be a topology generated by -neighborhoods and ideal . Then, -lower and -upper approximations and -accuracy of a subset are defined, respectively, for each as(1), where represents interior of w.r.t. (2), where represents closure of w.r.t. (3), where Henceforth, is with ideal on and denoted by .
Several fundamental properties of and are listed in the next proposition.
Proposition 3. Let be subsets of . Then, the following properties hold for each .(1)(2)(3)If , then (4)(5)(6)(7)(8)(9)(10)(11)If , then (12)(13)(14)(15)(16)
According to Example 2, Table 3 explains generalized rough approximations based on various topologies generated from -neighbourhoods via ideals.
Table 4 demonstrates that the accuracy measure for is the highest from the cases , and the accuracy measures for are the highest from the case . However, we can find another example illustrating that the accuracy measure for is the highest from the case .
In the following remark, the inclusion relation of parts 3, 5, 7, 11, 13, and 15 in Proposition 3 cannot be replaced by the equality relation.
Remark 1. Example 2 and Table 3 show that(1)If and , then and , whereas .(2)If , , then and . Hence, .(3)If , then .(4)If and , then and . Hence, .(5)If , then .
Definition 11. Let be on . A subset of is called(1)Totally definable, if (2)Internally definable, if and (3)Externally definable, if and (4) rough, if and
Remark 2. Example 2 and Table 3 show that is a totally -definable set, is an internally -definable set, is an externally -definable set, and is a -rough set.
The following theorem presents the relationships between the current approximations given in Definition 10 and the previous one which was mentioned in Definition 8.
Theorem 9. Let be and . Then, for each ,(1)(2)(3)
Proof. It follows from the fact that for each .
According to Theorem 2, the topologies and coincide for each when is a symmetric and transitive relation. This means that our approximations given in Definition 10 and those given in [21] are identical under a symmetric and transitive relation. On the other hand, there is a limitation in our approaches as the next result demonstrates.
Theorem 10. Let be , such that is a reflexive relation. Then, for each .
Proof. Let . Then, . Since for all under a symmetric and transitive relation (item 2 of Theorem 2) and is an ideal, then . Hence, .
4. Classifications of Approximations and Regions in Terms of -Neighborhoods and Ideals
By using ideals and -neighborhoods, we deduce another method to define -lower and -upper approximations, -boundary region, and -accuracy measure of a subset . Then, these kinds of approximations are compared with those in Section 3.2. Also, we clarify the relationships among them with the aid of several examples and show their essential properties.
Definition 12. Let be a subset of . Then, the -lower and -upper approximations of a subset are defined as(1)(2)(3), It should be noted that when in Definition 12, then the present approximations and coincide with the previous ones and in Definition 6. So, the current work is considered as a generalization of Al-Shami et al. work [18]. Now, we shall study the properties of and as it is presented in the following results.
Proposition 4. Let be a subset of . Then, the following properties hold for each .(i) and (ii) and (iii) and (iv) and (v) and (vi) and
Proof. Straightforward.
Theorem 11. Let be subsets of . Then, the following properties hold for each .(i)(ii)If , then (iii)(iv)(v)If , then (vi)(vii)If , then (viii)(ix)(x)If , then
Proof. We prove only (i), (ii), (iii), (iv), and (v), and the rest of proof is similar.(i)For each , we have ; then, .(ii), then . Since , , i.e., . So, .(iii)It follows from (ii) that . Conversely, let . Then, and . Therefore, and . Thus, . Hence, , as required.(iv)Since iff , iff , iff , iff . Then, the result holds.(v)ObviousIn view of (ii) and (vii) of Theorem 11, the next corollary is obvious.
Corollary 1. Let be subsets of . Then, the following properties hold for each .(i)(ii)
Example 3. In view of example 1, Table 5 offers approximations for each .
According to Table 5, we construct Table 6 which represents the -accuracy measure for every subsets of .
To elucidate that the converses of the items (ii), (v), (vii), and (x) of Theorem 11 and Corollary 1 are not always credible, we display the next example.
Example 4. In view of Example 1 and Table 5, note the following.(i)For and , we have and . But, .(ii)For , . But, .(iii)For , we find , we have and . But, .(iv)For , . But, .(v)For and , we have , , and . Then, .(vi)For and , we have , , and . Then, .Note that some basic properties of rough sets with respect to -lower and -upper approximations may evaporate. In what follows, we mention those missing properties.(i)(ii)(iii)(iv)(v)(vi)The next example supports assertions of the above note.
Example 5. In view of Example 1 and Table 5, note the following.(i)If , then . Hence, .(ii).(iii)If , then and . Hence, .(iv)If , then . Hence, .(v).(vi)If , then and . Hence, .
Lemma 3. Let be two ideals on and . If , then the following statements hold for each :(1)(2)(3)
Proof. Straightforward.
The next theorem shows the relationships between the approximations and of the current approximations in Definition 6 and and of Definition 6 for each .
Theorem 12. Let and . Then, the following statements hold for each :(1)(2)(3)
Proof. Direct to prove.
To preserve some Pawalk’s properties which are loss in the approximations given in Definition 12, we formulate new approximations and accuracy measures in the following.
Definition 13. Let be a subset of . Then, the -lower and -upper approximations and -accuracy measure of a subset are(1)(2)(3), .
Remark 3. It should be noted that the current approximations and in Definition 13 have the same properties of the current approximations and , which are stated in Theorem 11 and Corollary 1. Additionally, it satisfies the following properties:(1)(2), .
5. Conclusion and Future Work
Rough set is a powerful mathematical to deal with uncertainty. Approximation operators are the core concepts in rough set content; they have topological properties similar to all/some properties of the interior and closure operators. Neighborhood systems are one of the followed methods to study rough approximations using topological interior and closure operators.
In this study, we have initiated different types of topologies from ideals and -neighborhoods induced from any binary relations. We have applied these topologies to study new kinds of approximations and accuracy measures. Then, we have compared between them and their counterparts induced directly from -neighborhoods and ideals. Also, we have illustrated the advantages of our approaches to obtain higher accuracy measures than those proposed in [18]. Some limitations of our approaches have been investigated.
In the upcoming works, we will study new types of topologies and approximations induced from other neighborhoods and ideals. Also, we will investigate the concepts and results presented, herein, in fuzzy and rough sets contents.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.