Topological Approaches for Rough Continuous Functions with Applications

Salama, A. S.; Mhemdi, A.; Elbarbary, O. G.; Al-shami, T. M.

doi:https://doi.org/10.1155/2021/5586187

Complexity

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Special Issue

Dynamic Analysis, Learning, and Robust Control of Complex Systems

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 5586187 | https://doi.org/10.1155/2021/5586187

Topological Approaches for Rough Continuous Functions with Applications

A. S. Salama,¹A. Mhemdi,²O. G. Elbarbary,¹and T. M. Al-shami³

Academic Editor: Ahmed Mostafa Khalil

Received17 Feb 2021

Revised06 Mar 2021

Accepted13 Mar 2021

Published30 Mar 2021

Abstract

In this paper, we purposed further study on rough functions and introduced some concepts based on it. We introduced and investigated the concepts of topological lower and upper approximations of near-open sets and studied their basic properties. We defined and studied new topological neighborhood approach of rough functions. We generalized rough functions to topological rough continuous functions by different topological structures. In addition, topological approximations of a function as a relation were defined and studied. Finally, we applied our approach of rough functions in finding the images of patient classification data using rough continuous functions.

1. Introduction

Many studies have appeared recently and dealt with generalizations of topological near-open sets [1, 2] and the possibility of using them in many life applications, including their use in data reduction and reaching some new decisions and conclusions. Rough set theory is a modern approach for reasoning about data [3–7]. This theory depends on a certain topological structure that achieved great success in many areas of real-life applications [8–14]. Now, the general topologists can say, “rough sets theory is a topological bridge from real-life problems to computer science” [15, 16].

Rough set theory was introduced as a novel approach to processing of incomplete data. Among the aims of the rough set theory is a description of imprecise concepts. Suppose we are given a finite nonempty set of elements, called universe. Each element of is characterized by a description, for example, a set of attribute values. In rough sets formulated by Pawlak, an equivalence relation on the universe of elements is determined based on their attribute values. In particular, this equivalence relation is initiated using the equality relation on the attribute values. Many real-world applications have both nominal and continuous attributes [17–19]. It was early recognized that standard rough set model based on the indiscernibility relation is well suited in the case of nominal attributes.

Several procedures were made to overcome limitations of this approach and many authors presented interesting extensions of the initial model (see, for example, [20–24]). It was noted that considering a similarity relation instead of an indiscernibility relation is quite relevant. A binary relation forming classes of objects, which are identical or at least not noticeably different in terms of the available description, can represent the similarities between objects [25–29]. More recent approaches of rough set with its applications can be found in [30–32]. Other rough set theory applications in computer science (field of information retrievals) using topological generalizations can be found in [33–40].

In this paper, we purpose further study on rough functions and introduce new concepts based on rough functions. In Section 2, we give more details regarding the fundamentals of near-open sets. The goal of Section 3 is to introduce the concepts of topological lower and upper approximations of near-open sets and discuss their basic properties. We spotlight on rough numbers in Section 4. We aim in Section 5 to define and study new topological neighborhood approach of rough functions. Section 6 is devoted to generalize the concept of rough function to topological rough function by using different topological structures. Topological approximations of a function as a relation are defined and studied in Section 7. In Section 8, we suggest some applications of rough functions to information systems and give some applications of them in data retrieval. Finally, conclusions of the work are given in Section 9.

2. Basic Concepts of Topological Near-Open Sets

In this part, we recall the definitions of some near-open subsets of a topological space which are useful in the sequel.

A subfamily of the power set of is called a topology if it contains as well as it is closed under arbitrary union and finite intersection. The pair is called a topological space; elements in are called open sets, and their complements are called closed sets.

For a subset of , , and denote respectively the closure, interior, and complement of in , respectively.

A subset of is called,(1)Semi-open (resp., pre-open, open) set if (resp., , ) and its complement is called a semi-closed (resp., pre-closed, closed) set if (resp., , ). A subset which is both semi-open and semi-closed is called semi-regular(2)Semi-pre-open set (or open set) if and it is called a semi-pre-closed set (or closed set) if (3)Regular-open set if and it is called a regular-closed set if (4)-closed set if , where

The α-closure (resp. semi-closure, semi-pre-closure) of a subset of is the intersection of all –closed (resp. semi-closed, semi-pre-closed) sets that contain and is denoted by (resp., , ). The semi-interior of , denoted by , is the union of all semi-open subsets of .

A subset of a topological space is called(1)Generalized closed set if whenever and .(2)Semi-generalized closed (briefly, sg-closed) set if whenever and is semi-open set. Its complement is called a -open set.(3)Generalized semi-closed set if whenever and .(4)-Generalized closed set if whenever and .(5)Generalized α-closed set if whenever and is α- open.(6)-closed set if whenever and is α- open.

3. Topological Near-Open Approach of Rough Approximations

In this section, we introduce and investigate the concepts of topological lower and upper approximations of near-open sets and study their basic properties.

Let be a topological space. If , then(1)Semi-lower approximation of , , where is the family of all semi-open sets in . If we replace the family of all semi-open sets given in (1) above by a family of all pre-open sets (resp., a family of all α- open sets , a family of all -open sets , a family of all regular-open sets , and a family of semi-regular-closed sets ), we obtain pre-lower approximation (resp., α-lower approximation, -lower approximation, regular-lower approximation, and semi-regular-lower approximation).(2)Semi-upper approximation of ,, where is the set of all semi-closed sets in .

If we replace the family of all semi-closed sets given in (2) above by a family of all pre-closed sets (resp., a family of all α-closed sets , a family of all -closed sets , a family of all regular-closed sets , and a family of semi-regular-open sets ), we obtain pre-upper approximation (resp., α-upper approximation, -upper approximation, regular-upper approximation, and semi-regular-upper approximation).

Motivation for topological rough set theory has come from the need to represent subsets of a universe in terms of topological classes of the topological base generated by the general binary relation defined on the universe. That base characterizes a topological space, called topological approximation space, . The topological classes of are also known as the topological granules, topological elementary sets, or topological blocks; we will use to denote the topological class containing . In the topological approximation space, we consider two operators and called the topological lower approximation and topological upper approximation of , respectively. Also, let denote the topological positive region of denotes the topological negative region of , and denotes the topological borderline region of .

The degree of topological completeness characterizes by the topological accuracy measure, in which represents the cardinality of set as follows:

We define here the semi-rough pairs as an example of topological rough sets and we study their properties. You can use any type of the abovementioned near-open sets as another example.

The semi-topological class on a topological approximation space is determined by . A subset is said to be semi-dense (semi-co-dense) if (). By semi-rough pair on , we mean any pair where satisfies the conditions: (Semi-1) is the semi-open set in . (Semi-2) is the semi-closed set in . (Semi-3) . (Semi-4) there is a subset such that(1),(2),(3).

Lemma 1. For any subset in the topological approximation space , the pair is a semi-rough pair on in which every semi-open set in is a semi-closed set.

Proof. Let and . Then, the conditions from (Semi-1) to (Semi-3) are directly satisfied. Now, we need to prove condition (Semi-4). Define , then we have(1)If is a semi-open set, hence that gives which is a contradiction; hence, is not contained in . Then, it must be which gives .(2)Since , , then . Then, we have .(3)Let , , this means that . If , then for every semi-open set and such that implies that and we have , then . If , then there is a semi-open set and . Now, is a semi-open set which contains , and , then there exists a point such that , hence ; therefore, , hence . Then, we have the result .

Lemma 2. For any semi-rough pair in in which every semi-open subset is semi-closed, there are subsets such that and .

Proof. Let be a semi-rough pair and let be any subset, satisfying condition (Semi-4). Define , then , hence . If is a semi-open set, then is another semi-open set contained in . Since , , then , and we have . Therefore, is a semi-open set contained in which means . Since , it follows that and this proves that .
Now, we have . Also, and hence . Then, we have .

Theorem 1. For any topological subspace of the topological approximation space , the function that defined by is bijection.

Proof. First, we will prove that the function is onto as follows: for any semi-rough pair in , then there exists such that . Second, for the proof that is one to one, if , then which implies to and and .

4. Topological Neighborhood Approach of Rough Continuity

Let and be two subsets of a universe , and let and be two approximation spaces, where and are binary relations on and , respectively. We define two subsets and of (also two subsets and of ) which are called right and left neighborhoods of an element . We define now two topologies on and on , respectively, using the intersection of the right and left neighborhoods and as follows:

The rough approximations using these topologies are defined as follows:

The function is called a rough function on if the image of each rough set in is rough in .

Namely, the function f is totally rough iff all subsets , such that , then in .

The function is possibly rough iff some subsets , such that , then in .

Finally, the function is exact iff all subsets , such that , then in .

The function is a topological rough, continuous function on as the following:(1)The function is topological, totally rough, continuous if for all subsets ; if , then in .(2)The function is topological, possibly rough, continuous if for some subsets ; if , then in .(3)Finally, the function is topological exact continuous if for all subsets ; if , then in .

Example 1. Let and be topological spaces, where and and , . Let be a map defined by and , then our results are given in Table 1.
Then, according to Table 1, the function is a topological totally rough continuous function.

Proposition 1. Let and be topological spaces and let be a function. The following are equivalent:(1) is rough continuous.(2)For every , .(3)For every , is a rough continuous at .(4)For every , .

Proof. We will use the sequence (3) implying (1) implying (4) implying (2) implying (3) to prove the equivalence of the proposition. (3) implying (1): suppose a nonempty open set , for a fixed point , we have . But since is rough continuous at , then there exists an open set such that and , then we have and ; this gives that is rough continuous. (1) implying (4): suppose that is rough continuous and let . Let . Let an open set such that . Then, by the definition of rough upper approximation . Let , then . Then, we have . Then, we have . (4) implying (2): fix a closed subset ; let ; we will prove that . But each subset is contained in its upper approximation, . Now, we will prove that . Let , then using (4), we have ; hence or . Then, we have . (2) implying (3): let and be an open set containing . Then, is a closed set and is a closed set in which does not contain the point . But . Then, there exists an open set containing such that , then . Then, is rough continuous at .

Theorem 2. Suppose that be a family of topologies defined on . Let be a rough continuous function for every where is a topological space. Then, is a rough continuous function with respect to the topology .

Proof. Let , then , since is a rough continuous function for every , then . Then, we have in , hence is a rough continuous function with respect to the topology .

Theorem 3. Let be a family of functions. Suppose that is the topology on generated by the class , then(1) is rough continuous for each .(2)If is the intersection of all topologies on such that is rough continuous for each ; then, .(3) is the coarser topology on which gives that is rough continuous for each .(4)The class is a sub-base of .(5)The function is rough continuous if and only if is rough continuous.

Proof. Part (1): for each function if then and . But , then , hence ; then, we have the result. Part (2): we can easily prove that , but the topology is generated by , then . Otherwise, is one of the topologies that make the functions which are rough continuous. Then, we have , hence . Part (3): it is obvious by proof of Part (2). Part (4): since any collection of subsets of is a sub-base of a topology on , then is a sub-base of the topology . Part (5): if the function is rough continuous, then all functions are rough continuous. Otherwise, let be rough continuous and let , then there exists a subset such that . But . Now, we have , then . Then, is rough continuous.

5. Minimal Neighborhood Approach for Rough Continuity

We generalize the concept of rough function to topological rough function by using topological structures. The topological spaces with rough sets are very useful in the field of digital topology which is widely applied in the image processing in computer sciences.

Let be a topological space and . Then, we define

which is called the minimal neighborhood containing the point with respect to the topology on . Let be a topological space, for any element ; we define the subset which is the closure of in .

If is a function between two topological spaces and , we define the functions by for every .

Let be a function, where and are topological spaces. The function is called a topological rough function on if and only if for every . Also, is a topological rough function on if for every point in .

Example 2. Let and be topological spaces, where and and , . Let be a map defined by and , thenThen, we haveAlso,Then, the function is not a topological rough function on and .
A function is said to be topological roughly continuous at the point if and only if , and it is topological roughly continuous on if it is topological roughly continuous at every point .

Example 3. Let be a function defined by and , where and withThen, is a topological rough function on andthen for every , and then is a topological rough continuous function on .

6. Topological Approximations of a Function as a Relation

The function is a relation from to when it satisfies the conditions:(i),(ii)If and , then .

If , we say is a function on . By this way, any function can completely be represented by its graph .

Let be any function, where and are approximation spaces, such that and are any binary relations on and , respectively. We define the relation such that is the blocks of . For the function, , we define the approximations

A function is said to be rough in the approximation space , where and are approximation spaces and , if ; otherwise, is an exact function.

Example 4. Let and be two universes; we define the function , by its graph . Consider the blocks of the binary relations and as follows:Then,Then, we haveTherefore, the function is a rough function such that .
When we have two approximation spaces defined by two equivalence relations, we have the following proposition that governs the product space.

Proposition 2. Let and be two arbitrary approximation spaces. Then, we have .

Proof. Suppose that , and , then we haveSuppose again that .
Then, we haveThen, we have the result as .
Let be any function, where and are arbitrary approximation spaces. We define the relation to be the graph of the function . The rough approximations of are defined as follows:Accordingly, the function is rough if ; otherwise, is an exact function. The pair is called a rough pair of relations.
The following theorems give the conditions on approximation spaces that give exact functions, one-to-one, surjective, and continuous functions.

Theorem 4. The function is an exact function for any selective approximation spaces and .

Proof. The selective approximation space property means that . Then, we have , which yields to that the function is an exact function.

Theorem 5. The function is one-to-one function for any selective approximation spaces and if and only if both and are one-to-one functions.

Proof. The proof is directly using the definition of selective approximation space and using the technology in Theorem 1.

Theorem 6. The function is a surjective function for any selective approximation space and if and only if both and are surjective functions.

Proof. One can prove the theorem using similar technique given in Theorem 1.

Theorem 7. The function is a continuous function for any selective approximation space and if and only if both and are continuous functions.

Proof. As in the technique used in Theorem 5, when we have two topological spaces, generated using two bases , where and are two approximation spaces, then we have the following proposition that governs the product topology.

Proposition 3. Let and be two arbitrary topological spaces. Then, we have .

Proof. Similar to the proof of Proposition 2, the rough pairs of relations satisfied the following two important theorems.

Theorem 8. For the quasidiscrete product topological space , if is a rough pair of relations, and is a subspace of such that is closed in , then is a relative rough pair of relations when .

Proof. The pair is a rough pair of relations in , if the following condition satisfied the following:(1) is an open relation in .(2) is a closed relation in .(3).(4)The relation contains a relation of such that and .Only we need to prove that is a rough pair of relations in ; the proof will end by(1)Since is an open relation in , and is a subspace of , then is an open relation in .(2)Since is a closed relation in , then there is an open relation , such that , then , but is an open relationship with , then is a closed relation in the subspace .(3)Since , then .(4)By selecting , then the relation contains the relation , and we need to prove the two subconditions:(a),(b)_.For the proof of Part (a) , suppose that , then there is an -open relation such that but , i.e., , but such that is an open relation in , then ; hence, , but , which gives contradiction; then, it must be .
For the proof of Part (b), .
Since is a rough pair in , then there is a relation , such that and ; since , we have .
Now, let , then and .
Now, if , then and .
Finally, if and and , hence and . Now, , then there is an open relation in such that and , but , then . But since is an open relation in , and is an open relation in , then is a closed relation in , hence is an open relation in , hence is an open relation containing , then . This yields to , i.e., , such that . Hence, , but , then . But we have, hence implies that . Then, , but , hence . But and is an open relation in that contains , then, hence.

Theorem 9. Let be a rough pair of relations in the product topological space , and let be a subspace of such that is any relation of . Then, there is a relation such that and .

Proof. We can define , then , and then , but is an open relation in , i.e., , hence .
Finally, for, since is a rough pair in , then , i.e., implies that . But is a closed relation, i.e., , hence .
For , let , then and , but since , then , hence, i.e., , hence , then . Then, we have .
Let be a product space. For any relation , define the subspace of . We define the equivalence relation on the power set by for any . The set is a partition of and any class is called a relative topological rough relations.

Theorem 10. For any product topological space and for any subspace of it, the function , defined by , is bijection, where is the set of all relative rough pairs.

Proof. The proof is directly by Theorems 5 and 6.
Let and be any two topological spaces, where and are any two bases for and . Then, we define the base of the topology .
We define the approximations for any subset :The function on is called a topological rough continuous function at the point if for all open sets . The function is topological rough continuous on if it is topological roughly continuous at every point of .

Example 5. Consider the topology on and the topology on . The bases and are of and , respectively.
We defined the function as follows:Then, we haveThen, for any point , we have , then all open sets containing areThen, the inverse function of these open sets isThen, the function is topological rough continuous at every point of .

7. Future Applications of Topological Rough Functions on Information Systems

In this section, we will define a function between two information systems and give all needed conditions for them. Functions of an information system can produce the reductions, and the core of this system by the projection of the system on subsystems. We will define the image of rough set using some types of these functions. Finally, we define the topological rough functions of information systems and study some of their properties.

The reader can review about information systems in [7, 18] to know about the structure and the types and the different methods of reduction of information systems.

Suppose an information system where is the set of objects of this system (patients, plants, etc.). is the condition attributes of these objects (temperature, muscle pain, etc.). is the expert decisions about the condition attribute that objects suffer from.

We define the projection (restriction) function , where is the power set of the condition attributes as follows:

Figure 1 gives an example for a projection function on information system. The core of such systems is given by taking the intersection of all these projection functions on that system.

The topological rough continuous functions of information systems can be defined as follows:

The function is topological roughly continuous at the object if , where . The function is topological roughly continuous on if it is topological roughly continuous for every object of .

By a discernibility matrix of information system , denoted , we will mean matrix defined as follows:, wheresuch that or belongs to the C-positive region of ; is the set of all conditions attributed that classify objects and into different classes; denotes that this case does not need to be considered.

The discernibility function of an information system is defined as follows.

For any object, , where is the disjunction of all variables , when and , when and , when .

Figure 2 gives an example for a discernibility function on information system.

According to Figure 2, the function transfers the system into the discernibility and the reduction of this system can be obtained as follows:

Then, we have

Accordingly, the system has two reductions, namely, and with core .

8. Predictions of Patients Classification Data Using Rough Continuous Functions

Our aim in this application problem, which will give in this section, is to find recommendations for patients that combine treatment and exercise by explaining the function of each symptom, whether positive or negative.

In this application problem, the decision according to the medical reports is the continuation of taking all medicine and doing medical tests. In fact, it is a painful decision. Our role is to analyze the medical data using the notion of reduction which will help us to determine which of the patients can stop taking medicine as well as expect the required period of time to do that.

The structure is the mathematical style of information system of our patients problem. The set is the system universe that we selected to be a set of five patients. The set is the set of attributes of these patients with respect to tests functions such as liver, kidney, and heart functions. The set is values of each attribute . Finally, is the information function such that .

For any subset , we define the relation ; for , we define the class as follows: , where .

The structure is a decision table, where is the set of decisions that represents for each patient if he needs surgery or enough drugs.

We define the relation of the decision attribute by

The class of this relation is . The set of all classes is. We define the set to be a reduct of , if and is minimal.

Basic data of five patients before the surgery are given in Table 2 (the decision system of patients). Each patient will measure these medical functions periodically every three months. After a period of time, we need to predict the results of the medical tests of patients at any time and accordingly they can stop drugs. Therefore, we defined the prediction function , where is the decision system of patients over time (dynamic decision system of patients).

Now, if we choose for the liver function attributes , the threshold , then . The topology generated by is given by

For kidney functions, we can choose for , then ,

For the heart efficiency attribute , we can choose , and then ,

For the decision attributes, we have , then the topology of decisions is .

The condition attributes are exactly three attributes, namely, . The numbers of nontrivial subsets of the set of all condition attributes are seven subsets, namely, .

Now, we will calculate the classes of the residue subsets by taking the intersections as follows: , with topology . , with topology . , with topology .

The covering class of universe using all condition attributes is given as follows: , with topology . Then, the system given in Table 2 has no topological reductase.

Now, we define the function by . Then, according the function , the image of Table 2 after a period of three months has no topological reduces and this function is one-to-one rough continuous function.

9. Conclusion and Future Work

The emergence of topology in the construction of some rough functions will be the bridge for many applications and will discover the hidden relations between data. Topological generalizations of the concept of rough functions open the way for connecting rough continuity with the area of near continuous functions.

Applications of topological rough functions of information systems open the door about the many transformations among different types of information systems such as multivalued and single-valued information systems.

Future applications of our approach in the computer can be as follows: In information retrieval fields, we can modify the query running online by defining a function that converted documents to weighted vectors of the words of that document. Then, we can extract the results of weights in a decision table that we can classify the documents according to the reduction of this table. The query is constructed by defining a Boolean function of all words of the reduction. Classification and summarization of documentation using topological functions of neighboring systems are defined in documents.

Finally, we will benefit from the applications given in [41–47] to investigate new practical applications using rough sets and soft sets.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

T. M. Al-shami, “Somewhere dense sets and -spaces,” Punjab University Journal of Mathematics, vol. 49, no. 2, pp. 101–111, 2017.
View at: Google Scholar
T. M. Al-shami and T. Noiri, “More notions and mappings via somewhere dense sets,” Afrika Matematika, vol. 30, no. 7-8, pp. 1011–1024, 2019.
View at: Publisher Site | Google Scholar
Z. a. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341–356, 1982.
View at: Publisher Site | Google Scholar
G. Liu and Y. Sai, “A comparison of two types of rough sets induced by coverings,” International Journal of Approximate Reasoning, vol. 50, no. 3, pp. 521–528, 2009.
View at: Publisher Site | Google Scholar
T.-J. Li, Y. Leung, and W.-X. Zhang, “Generalized fuzzy rough approximation operators based on fuzzy coverings,” International Journal of Approximate Reasoning, vol. 48, no. 3, pp. 836–856, 2008.
View at: Publisher Site | Google Scholar
C. Degang, Y. Wenxia, and F. Li, “Measures of general fuzzy rough sets on a probabilistic space,” Information Sciences, vol. 178, pp. 3177–3187, 2008.
View at: Google Scholar
W. Zhu, “Topological approaches to covering rough sets,” Information Sciences, vol. 177, no. 6, pp. 1499–1508, 2007.
View at: Publisher Site | Google Scholar
A. Skowron and J. Stepaniuk, “Tolerance approximation spaces,” Fundamenta Informaticae, vol. 27, no. 2, pp. 245–253, 1996.
View at: Publisher Site | Google Scholar
R. Slowinski and D. Vanderpooten, “A generalized definition of rough approximations based on similarity,” IEEE Transactions on Knowledge and Data Engineering, vol. 12, no. 2, pp. 331–336, 2000.
View at: Publisher Site | Google Scholar
A. El-Sayed, “Abo-tabl, on links between rough sets and digital topology,” Applied Mathematics, vol. 5, pp. 941–948, 2014.
View at: Google Scholar
E. F. Lashin, A. M. Kozae, A. A. Abo Khadra, and T. Medhat, “Rough set theory for topological spaces,” International Journal of Approximate Reasoning, vol. 40, no. 1-2, pp. 35–43, 2005.
View at: Publisher Site | Google Scholar
K. Qin and Z. Pei, “On the topological properties of fuzzy rough sets,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 601–613, 2005.
View at: Publisher Site | Google Scholar
A. S. Salama, “Topological solution of missing attribute values problem in incomplete information tables,” Information Sciences, vol. 180, no. 5, pp. 631–639, 2010.
View at: Publisher Site | Google Scholar
A. Wiweger, “On topological rough sets,” Bulletin of the Polish Academy of Sciences Mathematics, vol. 37, pp. 89–93, 1989.
View at: Google Scholar
H. Yu and W.-r. Zhan, “On the topological properties of generalized rough sets,” Information Sciences, vol. 263, pp. 141–152, 2014.
View at: Publisher Site | Google Scholar
P. Srinivasan, M. E. Ruiz, D. H. Kraft, and J. Chen, “Vocabulary mining for information retrieval: rough sets and fuzzy sets,” Information Processing & Management, vol. 37, no. 1, pp. 15–38, 2001.
View at: Publisher Site | Google Scholar
H. M. Abu-Donia, A. A. Nasef, and E. A. Marei, Finite Information Systems, Applied Mathematics and Information Sciences, vol. 1, pp. 13–21, 2007.
S. Zhao and E. C. C. Tsang, “On fuzzy approximation operators in attribute reduction with fuzzy rough sets,” Information Sciences, vol. 178, no. 16, pp. 3163–3176, 2008.
View at: Publisher Site | Google Scholar
Z. Gong, B. Sun, and D. Chen, “Rough set theory for the interval-valued fuzzy information systems,” Information Sciences, vol. 178, no. 8, pp. 1968–1985, 2008.
View at: Publisher Site | Google Scholar
T. Herawan, “Roughness of sets involving dependency of attributes in information systems,” International Journal of Software Engineering and Its Applications, vol. 9, no. 7, pp. 111–126, 2015.
View at: Publisher Site | Google Scholar
M. Kryszkiewicz, “Rough set approach to incomplete information systems,” Information Sciences, vol. 112, no. 1–4, pp. 39–49, 1998.
View at: Publisher Site | Google Scholar
H. M. Abu-Donia, “Comparison between different kinds of approximations by using a family of binary relations,” Knowledge-Based Systems, vol. 21, no. 8, pp. 911–919, 2008.
View at: Publisher Site | Google Scholar
K. Qin, J. Yang, and Z. Pei, “Generalized rough sets based on reflexive and transitive relations,” Information Sciences, vol. 178, no. 21, pp. 4138–4141, 2008.
View at: Publisher Site | Google Scholar
H. Zhang, Y. Ouyang, and Z. Wang, “Note on “Generalized rough sets based on reflexive and transitive relations,” Information Sciences, vol. 179, no. 4, pp. 471–473, 2009.
View at: Publisher Site | Google Scholar
W. Zhu, “Relationship between generalized rough sets based on binary relation and covering,” Information Sciences, vol. 179, no. 3, pp. 210–225, 2009.
View at: Publisher Site | Google Scholar
W. Zhu, “Generalized rough sets based on relations,” Information Sciences, vol. 177, no. 22, pp. 4997–5011, 2007.
View at: Publisher Site | Google Scholar
J. A. Pomykala, “Approximation operations in approximation space,” Bulletin of the Polish Academy of Sciences, vol. 9-10, pp. 653–662, 1987.
View at: Google Scholar
U. Wybraniec Skardowska, “On a generalization of approximation space,” Bulletin of the Polish Academy of Sciences, vol. 37, pp. 51–61, 1989.
View at: Google Scholar
Y. Y. Yao, “Relational interpretations of neighborhood operators and rough set approximation operators,” Information Sciences, vol. 111, no. 1–4, pp. 239–259, 1998.
View at: Publisher Site | Google Scholar
Z. Bonikowski, E. Bryniarski, and U. Wybraniec-Skardowska, “Extensions and intentions in the rough set theory,” Information Sciences, vol. 107, no. 1–4, pp. 149–167, 1998.
View at: Publisher Site | Google Scholar
A. S. Mashhour, “New types of topological spaces,” Publicationes Mathematicae, vol. 17, pp. 77–80, 1970.
View at: Google Scholar
A. Kandil, M. Yakout, and A. Zakaria, “On bipreordered approximation spaces,” Journal of Life Science, vol. 8, pp. 505–509, 2011.
View at: Google Scholar
A. Kandil, M. Yakout, and A. Zakaria, “Generalized rough sets via ideals,” Annals of Fuzzy Mathematics and Informatics, vol. 5, pp. 525–532, 2013.
View at: Google Scholar
A. Kandil, M. M. Yakout, and A. Zakaria, “New approaches of rough sets via ideals,” Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing, IGI Global, Hershey, PA, USA, 2016.
View at: Publisher Site | Google Scholar
O. G. El Barbary, A. S. Salama, and A. El Sayed, “Granular information retrieval using neighborhood systems,” Mathematical Methods in the Applied Sciences, vol. 41, pp. 5737–5753, 2018.
View at: Publisher Site | Google Scholar
A. S. Salama, “Generalizations of rough sets using two topological spaces with medical applications,” Information, vol. 19, no. 7A, pp. 2425–2440, 2016.
View at: Google Scholar
O. G. El Barbary and A. S. Salama, “Feature selection for document classification based on topology,” Egyptian Informatics Journal, vol. 19, no. 2, pp. 129–132, 2018.
View at: Google Scholar
A. S. Salama and O. G. El Barbary, “Topological approach to retrieve missing values in incomplete information systems,” Journal of the Egyptian Mathematical Society, vol. 25, no. 4, pp. 419–423, 2017.
View at: Publisher Site | Google Scholar
A. S. Salama, “Sequences of topological near open and near closed sets with rough applications,” Filomat, vol. 34, no. 1, pp. 51–58, 2020.
View at: Publisher Site | Google Scholar
A. S. Salama, “Bitopological approximation space with application to data reduction in multi-valued information systems,” Filomat, vol. 34, no. 1, pp. 99–110, 2020.
View at: Publisher Site | Google Scholar
T. M. Al-shami, “Bipolar soft sets: relations between them and ordinary points and their applications,” Complexity, vol. 2021, Article ID 6621854, 14 pages, 2021.
View at: Publisher Site | Google Scholar
T. M. Al-shami, “Compactness on soft topological ordered spaces and its application on the information system,” Journal of Mathematics, vol. 2021, Article ID 6699092, 12 pages, 2021.
View at: Publisher Site | Google Scholar
T. M. Al-shami, “On soft separation axioms and their applications on decision-making problem,” Mathematical Problems in Engineering, vol. 2021, Article ID 8876978, 12 pages, 2021.
View at: Publisher Site | Google Scholar
T. M. Al-shami, “Soft separation axioms and fixed soft points using soft semiopen sets,” Journal of Applied Mathematics, vol. 2020, Article ID 1746103, 11 pages, 2020.
View at: Publisher Site | Google Scholar
T. M. Al-shami and M. E. El-Shafei, “Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone,” Soft Computing, vol. 24, no. 7, pp. 5377–5387, 2020.
View at: Publisher Site | Google Scholar
M. E. El-Shafei and T. M. Al-shami, “Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem,” Computational and Applied Mathematics, vol. 39, no. 3, p. 138, 2020.
View at: Publisher Site | Google Scholar
T. M. Al-shami and M. E. El-Shafei, “T-soft equality relation,” Turkish Journal of Mathematics, vol. 44, no. 4, pp. 1427–1441, 2020.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 A. S. Salama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

938

Downloads

1016

Citations