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On Some Topological Properties of Multigranular Rough Sets

R. Raghavan and B. K. Tripathy

Rough set theory was introduced by Pawlak [5] as a model to capture impreciseness in data and since then it has been established to be a very efficient tool for this purpose. The definition of basic rough sets depends upon a single equivalence relation defined on the universe or several equivalence relations taken one each taken at a time. There have been several extensions of the basic rough sets introduced since then in the literature. Rough set model based on tolerance relation ([1]) one of several such extensions. In the view of granular computing, classical rough set theory is researched by a single granulation. The basic rough set model has been extended to rough set model based on multi-granulations (MGRS) in [10], where the set approximations are defined by using multi-equivalences on the universe and their properties were investigated. Topological properties of rough sets introduced by Pawlak in terms of their types was recently studied by Tripathy and Mitra [15] to find the types of the union and intersection of such sets and also complement of one such set. In this paper we extend these results to the multi granulation context. The rough set model based on tolerance relations was also extended to the multi granulation context in [10, 11, 12] by introducing incomplete rough set model based on multi-granulations. Since the basic properties of both types of rough sets based on multi granulation are identical, our findings are also true for both complete and incomplete rough set models based upon multi granulation.