On Topologies Defined by Binary Relations in Rough Sets

Mathematical Problems in Engineering

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 j -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.

Rough Set Theory propsed by Zdzilaw Pawlak as a mathematical tool for approaching imprecise knowledge and it has many applications in Datamining, Machine Learning,Morphology, Control Theory and Medicine etc., The basic concept of Rough Set Theory depends on equivalence relation. In this paper we introduce topologies into the realm of Rough Sets and investigate their properties.Mathematical properties of Rough Set Theory can be constructed from fragments of Algebra, Logical Implications and Topology. We aim at presenting a properties of Rough Set Theory by Topological Methods

Journal of Mathematics

Neighbourhoods are one of the important topics in topology that relies on two types of neighbours of a point and has many applications in the graph theory and in the sciences and medical sciences. In this work, some types of bi-j-neighbourhoods on generalized bicovering approximation space are introduced. Moreover, some kinds of bicovering rough sets using relations are presented. Some properties of these new types of covering are discussed. Pawlaks’ properties are studied in the case of bicovering approximation space. More properties on different bi-neighbourhoods such as bi-j-neighbourhoods, complementary bi-j-neighbourhoods, and bi-j-adhesions are investigated. A comparison between these new types of bi -neighbourhoods and bi -covering is presented with the help of some counterexamples. Finally, we give an application of our results in the rheumatic fever data information by generated topologies.

This paper can be viewed as a generalization of Pawlak approximation space using general topological structure. Our approach depends on a general topology generated by binary relation. The introduced technique is useful because the concepts and the properties of the generated topology are applied on rough set theory and this open the way for more topological applications in rough context. Several properties and examples are provided.

Symmetry

In rough set theory, the multiplicity of methods of calculating neighborhood systems is very useful to calculate the measures of accuracy and roughness. In line with this research direction, in this article we present novel kinds of rough neighborhood systems inspired by the system of maximal neighborhood systems. We benefit from the symmetry between rough approximations (lower and upper) and topological operators (interior and closure) to structure the current generalized rough approximation spaces. First, we display two novel types of rough set models produced by maximal neighborhoods, namely, type 2 mξ-neighborhood and type 3 mξ-neighborhood rough models. We investigate their master properties and show the relationships between them as well as their relationship with some foregoing ones. Then, we apply the idea of adhesion neighborhoods to introduce three additional rough set models, namely, type 4 mξ-adhesion, type 5 mξ-adhesion and type 6 mξ-adhesion neighborhood rough models. ...

Rough Set Theory (RST) is a mathematical formalism for representing uncertainty that can be considered an extension of the classical set theory. It has been used in many different research areas, including those related to inductive machine learning and reduction of knowledge in knowledge-based systems. One important concept related to RST is that of a rough relation. This paper rewrites some properties of rough relations found in the literature, proving their validity.

International Journal of Approximate Reasoning 40 (2005) 35–43, 2005

The topology induced by binary relations is used to generalize the basic rough set concepts. The suggested topological structure opens up the way for applying rich amount of topological facts and methods in the process of granular computing, in particular, the notion of topological membership functions is introduced that integrates the concept of rough and fuzzy sets.
2005 Elsevier Inc. All rights reserved.

The present work is a study of the properties of the product rough approximation space determined by two rough approximation spaces. It is found that the projection functions are continuous with respect to the induced topologies on the product space and the individual spaces. Further, the product rough topology is defined as the topology of rough sets generated by the family of the products of the rough open sets in the rough topologies on the individual spaces. As in the case of general topological spaces, the product rough topology is found to be the smallest topology which makes the projection functions rough continuous. The rough interior and rough closure with respect to the rough product topology are also presented.

International Journal of Machine Learning and Cybernetics, 2016

The purpose of this paper is to investigate the relationship between multiset topology (M-topology) and generalized rough multiset induced by multiset relation. The notion of transitive closure of a relation is proposed in multiset context. The notable results we establish here are (1) any serial multiset relation induce an M-topology (2) any reflexive multiset relation R and its transitive closure t(R) induce the same M-topology, i.e., sðRÞ ¼ sðtðRÞÞ. The interior and closure operators of the M-topology sðRÞ is obtained and discuss their relationship with multiset relation R, when R is a reflexive multiset relation. On the other part, we consider the case when R is a tolerance multiset relation.

In this paper, we generalized the notions of rough set concepts using two topological structures generated by general binary relation defined on the universe of discourse. New types of topological rough sets are initiated and studied using new types of topological sets. Some properties of topological rough approximations are studied by many propositions.