Textures and dynamic relational systems
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ORIGINAL ARTICLE
Textures and dynamic relational systems Ays¸ egu¨l Altay Ug˘ur1
Received: 10 September 2015 / Accepted: 30 October 2016 Ó Springer-Verlag Berlin Heidelberg 2016
Abstract Dynamic relational systems have different forms in literature of rough set theory. They are divided into two parts by Pagliani as synchronic and diachronic dynamics. Synchronic dynamic case is related to the presence of a multi-sources. In the case of diachronic dynamics, it is supposed that changes occur in time. These changes are related to new objects, attributes or attribute values entered into the system. Here, we consider the dynamic systems which are related to the both of these types of dynamics. In a recent paper, the author studied on the results about reduct, definability and quasi-uniformity in dynamic relational systems. Here, the natural generalizations of these results are given. In particular, it is proved that weak and strong definabilities are preserved under pre-images with respect to direlation preserving difunctions between textural dynamic relational systems. Further, the connections between definable sets and the ditopology are determined by ditopologies of reflexive direlations. Then it is given some basic results on ditopologies induced by direlations and direlational quasi-uniformities. Morover, it is discussed on the connections between textural approximation spaces and direlational-quasi uniformities. Keywords Definability direlation ditopology dynamic relational system rough set textures
& Ays¸ egu¨l Altay Ug˘ur [email protected] 1
Department of Secondary Science and Mathematics Education, Hacettepe University, Beytepe, 06532 Ankara, Turkey
1 Introduction Rough set theory, proposed by Pawlak [32], is a powerful mathematical tool for dealing with vagueness and granularity in information systems. It is known that an information system is a quadruple (U, A, V, f) where U is a nonempty set of objects; S A is a nonempty set of attributes; V ¼ a2A Va and Va is a domain of attribute of a; f : U A ! V is an information function such that f ðx; aÞ 2 Va for every x 2 U; a 2 A. For any subset B A, the set INDðBÞ ¼ fðx; yÞ j 8a 2 B; f ðx; aÞ ¼ f ðy; aÞg is an equivalence relation on U. Note that the information system (U, A, V, f) can be represented by the relation IND(A). More generally, for any equivalence relation r on a universe U, the pair (U, r) can be regarded as an abstraction of an information system. In rough set theory, the pair (U, r) is called an approximation space. In fact, equivalence relations are too restrictive for real life applications. Recently, many papers on rough sets show that non-equivalence relations are more convenient for the fields such as machine learning, granular computing and data analysis [34–36, 44, 45]. To study the fundamental properties of approximation spaces, many authors consider an approximation space (U, r) where r is non-equivalence relation. Now consider the sets rs ðxÞ ¼ fy j ðx; yÞ 2 rg and rp ðxÞ ¼ fy j ðy; xÞ 2 rg for any x 2 U. The set r
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