Abstract
Topology is a beneficial structure to study the approximation operators in the rough set theory. In this work, we first introduce six new types of neighborhoods with respect to finite binary relations. We study their main properties and show under what conditions they are equivalent. Then we applied these types of neighborhoods to initiate some topological spaces that are utilized to define new types of rough set models. We compare these models and prove that the best accuracy measures are obtained in the cases of and . Also, we illustrate that our approaches are better than those defined under one arbitrary relation. To improve rough sets’ accuracy, we define some topological spaces using the idea of ideals. With the help of examples, we demonstrate that our methods are better than some methods studied in some published literature. Finally, we give a real-life application showing the merits of the approaches followed in this manuscript.
1. Introduction and Preliminaries
Information science is an area that is mostly interested in analysis, combination, sorting, storage, retrieval, and protection of information. On occasion, knowledge in information science is constructed in an imperfect or vague manner and has a level of granularity.
As one of the treatment methods of these issues of the knowledge system, rough set theory (briefly, RS theory), created by Pawlak [1], has been developed. The main purpose of RS methods is upgrading the approximation issues, which were established to reduce the boundaries and raise the degree of accuracy. RS theory has involved several topological notions; approximation operators have many characterizations of interior and closure operators that are topological operators derived from equivalence relations defined on the universe. Thus, the study of RS by using the concept of topology is helpful to analyze many issues in reality (see [2, 3]).
However, equivalence relations in the classical RS theory have some restrictions in generating neighborhood bases. To overcome these restrictions, the authors interested in uncertainty have studied the classical RS theory with respect to different types of relations [4], covering rough sets [5], and constructed different topological structures [6, 7]. Kondo [8] showed that a reflexive relation produces a topology. The authors of [9, 10] investigated the features of rough sets under different kinds of binary relations. Salama [11] explored some topological properties of rough sets and showed that rough sets and topological rough sets constitute a consistency base for data mining. Al-shami [12] improved the accuracy values of subsets using the concept of somewhere dense sets.
As yet, concerted studies of the rough set theory and topology are demanded. The idea of neighborhood systems is extracted from the geometrical concept of “near,” and it is inherent in the theory of topological spaces. By considerable researches in the discussion of RS, neighborhood systems have been applied. Yao [13] examined the RS using neighborhood systems for explicating granules. Abd El-Monsef et al. [14] employed the notion of “-neighborhood space” (-NS) to generalize the classical RS theory by utilizing various general topologies deduced from binary relations. Mareay [15] defined the concepts of core neighborhoods and applied them to initiate topological approximations. Al-shami et al. [16] initiated different rough set models using -neighborhoods.
Also, as a generalization of RS methods, Abu-Donia [17] exploited -neighborhoods to characterize new kinds of approximations of any set with respect to a finite family of binary relations . Then he [18] improved these approximations using -neighborhoods. Recently, Al-shami et al. [19] have defined a new system of neighborhoods called -neighborhoods and applied it to establish new rough and topological approximations. They compared them and showed that the accuracy measure obtained from rough approximations is better than their counterparts obtained from topological approximations. Also, Al-shami [20] has established a new family of neighborhood systems called -neighborhoods. He applied to protect medical staff from new coronavirus (COVID-19). Recently, Al-shami and Ciucci [21] have defined a new class of neighborhood systems namely “subset neighborhood.” They have explored their main properties and investigated their applications to handle some diseases.
The interaction between rough set theory and topology began by Skowron [22] and Wiweger [23]. Ideal structure, which was initiated by Kuratowski [24], is defined as a nonempty collection of subsets of a universe that is closed under finite union and subsets. The concept of an ideal topological space consists of topology and ideal defined on the same universe, and they are independent of each other. Using ideal topological spaces as a new technique to generate new rough approximations was introduced by [25]. After that, some studies were conducted on ideal rough approximations; see, for example, [26, 27]. Recently, Hosny et al. [28] have applied -neighborhoods and ideals to initiate new types of rough set models.
By using ideals notion and topologies generated by -neighborhoods with respect to the finite family of binary relations on a universe, our suggested approach will deal with the gap between topologies concepts and their applications in various fields of real life. We successfully applied our approaches to reduce the boundary region and increase the accuracy value, which is the essential target of RS theory.
In the rest of this section, we recall some concepts that help understand this manuscript well.
Definition 1. (see [20]) A binary relation on a nonempty set is said to be as follows:(i)Reflexive if for each (ii)Symmetric if (iii)Antisymmetric if whenever and (iv)Transitive if whenever and (v)Preorder (or quasiorder) if it is reflexive and transitive(vi)Equivalence (resp. partial order) if it is symmetric (resp. antisymmetric), reflexive and transitive(vii)Diagonal if (viii)Comparable if or for each
Definition 2. (see [20]). Let be a binary relation on a nonempty finite set (universe) and . The -neighborhoods of (symbolized by are defined as follows:(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)
Definition 3. A class of subsets of is closed under finite intersection, and the arbitrary union is called a topology. We call an ordered pair a topological space.
Definition 4. For a set in , the interior of (denoted by ) is the union of all open subsets that are contained in ; the closure of (denoted by ) is the intersection of all closed subsets containing .
2. -Neighborhoods Based on a Finite Number of Binary Relations and the Topologies Inferred from Them
In this part, we introduce and explore the -neighborhood space that depends on a finite number of arbitrary binary relations. Then we apply them to generate eight diverse topologies and investigate the relationships between them with help of examples. Also, we exploit these topologies to initiate lower and upper (rough) approximations. Comparisons between the accuracy of these types are attained, as well as comparisons that show our methods better in terms of accuracy measure than those given in [14] are presented.
Henceforth, for all the following results, we will deal with all the values of , , unless otherwise noted.
Definition 5. Let be a finite family of binary relations on . The -neighborhoods of with respect to (briefly, ) are defined as follows:(1)-neighborhood: . Equivalently, .(2)-neighborhood: . Equivalently, .(3)-neighborhood: .(4)-neighborhood: .(5)-neighborhood: (6)-neighborhood: (7)-neighborhood: .(8)-neighborhood: .
Remark 1. The concepts of -neighborhood and -neighborhood were introduced and studied in [17, 18], respectively.
Definition 6. Let be a finite family of binary relations on a universe and : be a mapping that assigns for each in its in . Then is called a -neighborhood space (briefly, ), .
Now, we investigate the main properties of under some types of binary relations.
Proposition 1. Let be a such that is reflexive for each . Then for each .
Proof. If is reflexive for each , then and . Hence, and .
Proposition 2. Let be a such that is preorder for each . Then for each .
Proof. It follows from the above proposition that for each . Conversely, let . Then there exists such that and . Suppose that . Since is transitive for each , then ; this is a contradiction. Therefore, , and consequently, . Thus, . Similarly, one can prove that . Hence, and .
Proposition 3. Let be a such that is reflexive and antisymmetric for each . Then(i)If , then and (ii) for each (iii) for all in the case of
Proof. (i)Let . Then for each . Since is antisymmetric for each , then does not hold for each . Therefore, and .(ii)Since is reflexive for each , then . Now, consider . Then and . Therefore, and for each . Since is antisymmetric for each , we obtain . Thus, . Since is reflexive for each , and , as required.(iii)For . Let . By hypothesis, is reflexive for each , we obtain and . By assumption, and . It follows from the antisymmetric of for each that .. Let . It follows from the reflexivity of for each that and . By assumption, and . Consequently, and . Thus, and . Since is antisymmetric for each , we obtain .
Proposition 4. Let be a . Then the following results hold:(i)The comparable of for each implies that or for all (ii)The symmetric of for each implies that and
Proof. Straightforward.
Proposition 5. Let be a such that is transitive and symmetric for all . If the intersection of and is nonempty, then and are equal for all .
Proof. We prove the result for . One can prove the other cases in a similar way.
Since , there is such that and . Since is symmetric for each , , and since is transitive for each , . Now, let . Then . Since , . Hence, . Following the similar argument, we find .
Proposition 6. Consider as a , where is reflexive and antisymmetric. Then, for , a class represents a partition for .
Proof. We suffice by proving case of .
The reflexivity of for each implies that . Consider . Suppose that . Then there is satisfying and . This leads to and . Since is antisymmetric for each , and . This is a contradiction with our assumption. Hence, for every .
Proposition 7. Let be a , where a relation is a partial order for each . The next results hold for the smallest element .(i)(ii)
Proof. (i)Let be the smallest element. Then for each . This implies that . Consequently, . To show that , let . By hypothesis, we find . That is, represents the only right neighborhood of . Thus, . As a direct result, we have .(ii)Similar to (i).
Proposition 8. Let be a , where a relation is partial order for each . The next results hold for the largest element .(i)(ii)
Proof. The proofs (i) and (ii) are similar to that of Proposition 7.
In the following result, we initiate a topology from -neighborhoods.
Theorem 1. Let be a , . Then the collection is a topology on .
Proof. Obviously, and belong to . Let for each . Let . Then there exists such that . Therefore, . Thus, belongs to , as required. Now, let . Then, for each , we have and . Consequently, . Hence, we obtain the desired result.
Now, one may study the relations between the topologies , , which generated from .
Proposition 9. Let be a , . Then the next properties hold true:(1)(2)(3)(4)
Proof. To prove 1, let . Then , . Therefore, , . Thus, and , . Hence, , as required. To prove 2, let . Then , , or , . Since or , . Hence, .One can prove the other cases similarly.
Example 1. On , we have information systems specified as follows:
,
such that is the identity relation on .
Therefore, we obtain the following:
, , , , . .
Remark 2. It can be seen from Example 1 that the eight topologies [2]– are different from each other.
Example 2. On , we have information systems specified as follows:,. Then , and.
Remark 3. , are not necessarily comparable, for any . Suppose , then(1)Example 1 illustrates (2)Example 2 shows that
Proposition 10. Let be a , . Then the topologies and are dual.
Proof. To prove the duality between , , we shall prove that if , that is,
, , .
Suppose , . If , then . Let for each . Then there exists and , that is, and , for each . So . implies that . Hence, . It is a contradiction, so . By the same manner, we can prove the other side.
Remark 4. In view of Example 1, the topologies , are not dual.
Definition 7. Let be a , . A set is called a open set if , and the complement of a open set is called a closed set. The family of all closed sets of a is defined by , where is the complement of .
Definition 8. Let be a , . If , then the lower, upper approximations, boundary regions, and accuracy of are defined, respectively, as follows:(1), where represents the interior points of with respect to (2), where represents the closure points of with respect to (3)(4), where
Definition 9. Let be a , . A set is called definable set if . Otherwise, is called rough set.
In the following example, we make a comparison between the approximations and accuracy measure of any subset of for each and . The following example shows that we obtain the best approximations and accuracy measure in the cases of and ; see Tables 1 and 2.
Example 3. On , we have information systems specified as follows:
, such that is an identity relation on , , , , , , .
Remark 5. In view of Table 2 and according to Example 3, one may notice the following:(1)For , is the preferable accuracy, and the best method for building the approximations of sets is specified by utilizing the topology [2]–(2)The accuracy measure of any set , , coincides with the accuracy measure due to [17] when , is a reflexive relation and
Remark 6. In view of Table 2 and according to Example 3, one may notice the following:(1), (2), (3), (4) is the preferable accuracy, and the best method for building the approximations of sets is specified by utilizing the topology [2]–(5)The accuracy measure of any set , , coincides with the accuracy measure due to [18] when , is a reflexive relation and .
Remark 7. According to Example 3, and are definable sets, and is rough set for all .
One of the obtained merits of using more than one relations is the improvement of the accuracy measures of a subset. To demonstrate this matter, we compare the accuracy measure obtained using two arbitrary binary relations and accuracy measure obtained using one arbitrary binary relation in the case of . In fact, this proves that our approach is better than that given in [14]; see the next example.
Example 4. On , we have information systems specified as follows:
, . Therefore, , , .
It can be seen from Table 3 that the rows with green color show that and .
3. ℐj-Lower and Upper Approximations
This part aims to generalize the topologies generated by , utilizing the idea of ideals. After we show the relationships between the new topologies, we apply them to define new rough approximations. We prove that our new techniques produce higher accuracy measures than those given in [17, 18, 26]. Some practical examples are provided.
Theorem 2. Let be an , . If , then the collection is a topology on .
Proof. Suppose , . Let , then there exists such that . Hence, . Since , then , that is, .
Suppose , . Let , then and . According to properties of , . Hence, . It follows that .
Easily, , . Consequently, is a topology on .
Lemma 1. Let be two ideals on a , . If , then .
Proof. Straightforward.
Proposition 11. Let be an , . Then the following statements hold:(1)(2)(3)(4)
Proof. We prove only (1), and the other cases can be proved similarly.
Let . Then , . Hence, , . So and , . Consequently, .
Example 5. In Example 3, suppose that . Then,,,.
We note that the topologies , are not dual. Also, the four topologies are different from each other.
Example 6. On , suppose that is an ideal and are two binary relations on defined as follows:,.Hence, ,,,.
Remark 8. In view of Example 1.5,(1) and (2) and (3)
Remark 9. and are not necessarily comparable, for any . Suppose , then(1)Example 6 illustrates (2)In Example 2, if , then
Theorem 3. Let be an , . If , then .
Proof. Let . Then , . So , . Hence, .
Using the chemical application mentioned in article [29], we make a comparison among the and with respect to reflexive relations.
Example 7. Let be five amino acids (AAs). The (AAs) are characterized in terms of five attributes: = PIE, = surface area, = molecular refractivity, = the side-chain polarity, and = molecular volume. See Table 2 of [29], which displays all quantitative attributes of five AAs.The five reflexive relations on defined as: , where symbolizes the standard deviation of the quantitative attributes , . The -neighborhoods for all elements of with respect to the relations , are shown in Table 3 of [29].So one may notice the following:, , , , .Hence, [5]–. If , then [5]–. Hence, [5]–.
Remark 10. In Example 6, for any arbitrary binary relations , then [2]–, and . Hence, .
Definition 10. Let be an , . A set of 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
By using ideals, we present eight methods for approximating rough sets via interior , closure , of the topologies , , which will be contributory in decision-making.
Definition 11. Let be an , . If , then the lower, upper approximations, boundary regions, and accuracy of a set are defined, respectively, as follows:(1)(2),(3)(4), where
Example 8. In Example 4, suppose that . Then,,,,.
Remark 11. In view of Example 8 and Table 4, for any arbitrary binary relations , the accuracy, , depending on Definition 11 is better than the accuracy measure due to [17, 18], , , respectively, in Example 4 and Table 3.
Several fundamental properties of the , operators in the next proposition are listed. Properties , j = 1, 2, …, 10, display that approximate operators , are dual to each other. By using and behaviors, the proof of the next proposition is understandable.
Proposition 12. Let be an , . If , then the following properties hold:(1) (2) (3) If , then (4) (5) (6) (7) (8) (9) (10) = (11) If , then (12) (13) (14) (15) (16)
Remark 12. Example 6 shows that the converse of does not hold; if and , then , but .If and , then . Consequently, the converse of is not true.If and , then . Consequently, the converse of is not true.If and , then the converse of is not true.
According to Theorem 3, the proof of the following theorem is evident.
Theorem 4. Let be an , . If , then ; the following statements hold:(1)(2)
Lemma 2. Let be two ideals on a , . If , then the following statements hold:(1)(2)(3)
Proof. Straightforward.
Example 9 offers the comparison between Definition 9 and Definition 6, if each , is a reflexive relation on .
Example 9. In Example 3, if , then,,.
Remark 13. It is clear that from Example 3, and .
Remark 14. In Example 9 and Table 5:(1) and (2) and (3) and
The following propositions are obvious, and the proof is omitted.
Proposition 13. Let be an , . If , then ; the following statements hold:(1)(2)(3)(4)
Proposition 14. Let be an , . If , then ; the following statements hold:(1)(2)(3)(4)
Corollary 1. Let be an , . Then and .
Definition 12. Let be an , , . A subset of is called as follows:(1)Totally, definable, if (2)Internally, definable, if and (3)Externally, definable, if and (4) rough set, if and
Corollary 2. Let be an , . Then ; the following statements hold:(1)Every definable subset in is definable(2)Every rough subset in is rough
Remark 15. Example 9 shows that the converse of parts of Corollary 2 is not necessarily true.(1)If , then is definable, but it is not definable (see Tables 2 and 5)(2)If , then is rough, but it is not rough (see Tables 2 and 5)
Remark 16. In view of Example 9 and Table 5, we have the following:(1) is a totally definable set(2) is an rough set(3) is an internally definable set(4) is an externally definable set
Corollary 3. Let be an , . Then ; the following statements hold:(1)(2)(3)(4)
In Example 10, we show that the current method in Definition 11 is more accurate in comparison to Hosny’s method [26], for any .
Example 10. Continued from Example 4. Table 6 shows the comparison between the current method in Definition 11 and Hosny’s method [26], for any .If , then,,.
4. Network Devices Application
Network devices or networking hardware (network interface cards, hubs, switches, repeaters, bridges, routers, and gateways) are physical devices that are required for communication and interaction between hardware on a computer network. This section is devoted to introducing an applied example (see [30]) of the new method on the network topologies. This application presents a comparison between approaches with respect to and , .
Example 11. Let be a set of four network topologies and be the attributes of network topologies (see Table 7), where is a bus topology, is a ring topology, is a star topology, and is a mesh topology. = The method of transfer data = = , = Cable type = = , and = Bandwidth capacity = = .Consider Table 7: let , be arbitrary binary relations as follows:
, , .
Then we compute the approximations of attributes , and . = , = , = . Now, , , , .Therefore, [3]–.If = , then [3]–.If = , then [3]–.
From Table 8, we notice the following:(1)The accuracy of the approximations induced from for any ideal is more accurate than the accuracy of approximations induced from (2)If , then the accuracy of the approximations induced from is more accurate than the accuracy of the approximations induced from , (3)Every definable set with respect to is definable with respect to for any ideal , but the converse is not true(4)The approximations induced from for any ideal will help extract and discover the hidden information in data that were collected from real-life applications, which are very useful in decision-making
5. Conclusion and Future Work
In this manuscript, we have presented novel types of rough neighborhoods generated from a finite number of binary relations. We have scrutinized their main features and determined under what conditions some of these neighborhoods are equivalent. Then we have established topological spaces from these types of neighborhoods that we used to initiate novel rough set models. The best approximations and accuracy measures have been given in the cases of i and as the provided examples illustrated. Thereafter, we have applied the notion of ideals to initiate topological spaces finer than those induced from rough neighborhoods given in Section 2. Some comparisons have been given to show the importance of our approaches compared with some methods in the literature such as [14, 17, 18, 26]. Finally, we have displayed a practical application showing the merits of the approaches followed in this manuscript.
In upcoming works, we plan to benefit from the rough neighborhoods given in [12, 20] and the ideals to improve the approximations given herein and increase their accuracy values.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.