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An Approach of Proximity in Rough Set Theory

Tiwari Surabhi, Singh Pankaj Kumar

Fundamenta Informaticae

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2019

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Vol. 166, nr 3

251--271

EN

In this paper, we have constructed topological structures on rough sets by choosing the path of proximity relations on approximation spaces. So, by this virtue of purpose, we have used rough metric to define nearness concept between rough sets. Some basic results have been proved on this new nearness structure named as rough proximity. The study is well supported by examples. Finally, the theory is developed to construct the compactification of a rough proximity space.

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A New Proof on Embedding the Category of Proximity Spaces into the Category of Nearness Spaces

Yang Z.

Fundamenta Informaticae

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2008

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Vol. 88, nr 1-2

207-223

EN

This paper establishes an explicit embedding functor from the category of the proximity spaces into the category of nearness spaces. This is done by defining methods to constructively inducing a proximity structure with a given nearness structure and vis versa. Those induction methods provide useful tools to understand the inter-relations and to translate the result between the two types of spaces.

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Nearness of Objects: Extension of Approximation Space Model

Peters J.F., Skowron A.

Fundamenta Informaticae

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2007

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Vol. 79, nr 3-4

497-512

EN

The problem considered in this paper is the extension of an approximation space to include a nearness relation. Approximation spaces were introduced by Zdzisaw Pawlak during the early 1980s as frameworks for classifying objects by means of attributes. Pawlak introduced approximations as a means of approximating one set of objects with another set of objects using an indiscernibility relation that is based on a comparison between the feature values of objects. Until now, the focus has been on the overlap between sets. It is possible to introduce a nearness relation that can be used to determine the "nearness" of sets of objects that are possibly disjoint and, yet, qualitatively near to each other. Several members of a family of nearness relations are introduced in this article. The contribution of this article is the introduction of a nearness relation that makes it possible to extend Pawlak's model for an approximation space and to consider the extension of generalized approximations spaces.

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Contact Algebras and Region-based Theory of Space: Proximity Approach - II

Dimov G., Vakarelov D.

Fundamenta Informaticae

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2006

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Vol. 74, nr 2,3

251-282

EN

This paper is the second part of the paper [2]. Both of them are in the field of region-based (or Whitehedian) theory of space, which is an important subfield of Qualitative Spatial Reasoning (QSR). The paper can be considered also as an application of abstract algebra and topology to some problems arising and motivated in Theoretical Computer Science and QSR. In [2], different axiomatizations for region-based theory of space were given. The most general one was introduced under the name ``Contact Algebra". In this paper some categories defined in the language of contact algebras are introduced. It is shown that they are equivalent to the category of all semiregular T0-spaces and their continuous maps and to its full subcategories having as objects all regular (respectively, completely regular; compact; locally compact) Hausdorff spaces. An algorithm for a direct construction of all, up to homeomorphism, finite semiregular T0-spaces of rank n is found. An example of an RCC model which has no regular Hausdorff representation space is presented. The main method of investigation in both parts is a lattice-theoretic generalization of methods and constructions from the theory of proximity spaces. Proximity models for various kinds of contact algebras are given here. In this way, the paper can be regarded as a full realization of the proximity approach to the region-based theory of space.