Some Topological Approaches for Generalized Rough Sets via Ideals
TL;DR: A new rough set model derived from various topologies generated by means of the concepts ofhesion neighborhoods and ideals is defined and it is shown that these topologies are finer than those given in the previous ones under arbitrary binary relations.
Abstract: The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of - adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.
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TL;DR: In this paper , the authors introduce a topological method to produce new rough set models based on the idea of "somewhat open sets" which is one of the celebrated generalizations of open sets.
Abstract: Abstract In this paper, we introduce a topological method to produce new rough set models. This method is based on the idea of “somewhat open sets” which is one of the celebrated generalizations of open sets. We first generate some topologies from the different types of $$N_\rho $$ N ρ -neighborhoods. Then, we define new types of rough approximations and accuracy measures with respect to somewhat open and somewhat closed sets. We study their main properties and prove that the accuracy and roughness measures preserve the monotonic property. One of the unique properties of these approximations is the possibility of comparing between them. We also compare our approach with the previous ones, and show that it is more accurate than those induced from open, $$\alpha $$ α -open, and semi-open sets. Moreover, we examine the effectiveness of the followed method in a problem of Dengue fever. Finally, we discuss the strengths and limitations of our approach and propose some future work.
23 citations
TL;DR: Novel generalized rough set models using the concepts of “maximal left neighborhoods and ideals” are initiated to preserve almost all major properties of approximation operators with respect to the Pawlak model and enlarge the knowledge gotten from the information systems because they minimize the vagueness regions more than some previous models.
Abstract: Rough set theory was introduced by Pawlak, in 1982, as a methodology to discover structural relationships within imprecise and uncertain data. This theory has been generalized using the idea of neighborhood systems to be more efficient to get rid of uncertainty and deal with a wide scope of practical applications. Motivated by this idea, in this work, we initiate novel generalized rough set models using the concepts of “maximal left neighborhoods and ideals.” Their basic features are studied and the relationships between them are revealed. The main merits of these models, as we prove, are first to preserve almost all major properties of approximation operators with respect to the Pawlak model. Second, they keep the monotonic property, which leads to an efficient evaluation of the uncertainty in the data, and third, these models enlarge the knowledge gotten from the information systems because they minimize the vagueness regions more than some previous models. We complete this manuscript by applying the proposed approach to analyze educational data and illustrate its role to improve the obtained classifications of objects and show the great performance of the present approach against other ones. Elucidative examples that support the obtained results are provided.
10 citations
5 citations
TL;DR: In this paper , the authors introduce a debate about the structure of rough topological space based on the reflexive relation and define the relationships between the set of all topologies that satisfy the requirement of compactness.
Abstract: Recently, topological structures have emerged as one of the most popular rough sets (RS) research topics. It can be stated that it is a fundamental and significant subject in the theory of RS. This study introduces a debate about the structure of rough topological space based on the reflexive relation. To create the rough topological space, we use the representation of RS. We also look at the relationships between approximation operators, closure operators, and interior operators. Also, the relationship between topological space in the universe that is not limited or restricted to be ended, and RS induced by reflexive relations is investigated. Furthermore, we define the relationships between the set of all topologies that satisfy the requirement of compactness $ C_{2} $ and the set of all reflexive relations. Finally, we present a medical application that addresses the issue of dengue fever. The proposed structures are used to determine the impact factors for identifying dengue fever.
5 citations
TL;DR: In this article, the concept of rough sets has been introduced and defined based on the topologies generated by two ideals and the basic properties of these approximations are studied and compared to the previous ones and shown to be more general.
Abstract: Ideal is a fundamental concept in topological spaces and plays an important role in the study of topological problems. This motivated us to use two ideals to generate different topologies to take the advantage of the two ideals at the same time. Two ideals represent two opinions instead of one opinion which is very useful for using the insights of two groups of experts to study the problem and elicit decisions based on their common vision. Topology is a rich source for constructs that is helpful to enrich the original model of approximations spaces. Rough set theory has inbuilt topological concepts. Hence, the main purpose of this paper is to point out that the concept of rough sets has a purely topological aspects nature. To do so, new approximations spaces are introduced and defined based on the topologies generated by two ideals. The results in this paper show that the topological concepts can be a powerful method to study rough set models. The basic properties of these approximations are studied and compared to the previous ones and shown to be more general. The importance of the current paper is not only introducing a new kind of rough set based on bi-ideals, increasing the accuracy measure, and reducing the boundary region of the sets which is the main aim of rough set but also introducing a chemical application to explain the concepts.
5 citations
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Journal Article•
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TL;DR: The presented approach may be considered as an alternative to fuzzy sets theory and tolerance theory and some applications are outlined.
Abstract: W: Bulletin of the European Association for Theoretical Computer Science (EATCS), 24:94-109, 1984
1,379 citations
"Some Topological Approaches for Gen..." refers background or methods in this paper
...0e idea of rough set theory (RST) was first put forth by Pawlak [1], where imperfect information gives rise to indiscernibility of objects....
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...Hosny [22] replaced Pawlak’s approximations (lower and upper) by topological operators (interior and closure)....
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...By using any binary relation and without toting any supplement conditions as Proposition 6 claims, one might observe that our approximation approach satisfies all features of the classical theory of rough set [1]....
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...To deal with uncertainty issues, Pawlak [1] proposed a nonstatistical approach called rough set....
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TL;DR: Some fundamental properties of generalized rough sets induced by binary relations on algebras are considered and it is shown that any reflexive binary relation determines a topology.
289 citations
"Some Topological Approaches for Gen..." refers background in this paper
...So, a lot of ideas and articles have been introduced to generalizing the classical theory of RSs, for further specifics, see reference [2]....
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TL;DR: The notion of topological membership functions is introduced that integrates the concept of rough and fuzzy sets that opens up the way for applying rich amount ofTopological facts and methods in the process of granular computing.
235 citations
09 May 1999
TL;DR: The granulation structures used by both rough set theory and neighborhood systems, and the corresponding approximation structures, are studied.
Abstract: Granulation of a universe involves grouping of similar elements into granules. With granulated views, we deal with approximations of concepts, represented by subsets of the universe, in terms of granules. This paper examines the problem of approximations with respect to various granulations of the universe. The granulation structures used by both rough set theory and neighborhood systems, and the corresponding approximation structures, are studied.
92 citations
"Some Topological Approaches for Gen..." refers methods in this paper
...For interpretation of the granules, both Lin [3] and Yao [4] studied the RSs utilizing neighborhood systems....
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TL;DR: New types of neighborhoods called containment neighborhoods are introduced depending on the inclusion relations between j-neighborhoods under arbitrary binary relation and it is proved that a C j -accuracy measure is the highest in cases of j = i, 〈 i 〉 .
65 citations
"Some Topological Approaches for Gen..." refers background in this paper
...Al-shami [9] defined containment neighborhoods and applied to protect medical staff from infected diseases....
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