Topological Properties of Incomplete Multigranulation Based on Rough Fuzzy Sets

Abstract

AbstractThe definition of basic rough sets [3] depends upon either a single equivalence relation defined on a universe or several equivalence relations defined over the universe, taken one each taken at a time. In the view of granular computing, classical rough set theory is based upon single granulation. Extending this notion, a rough set model based on multi-granulations (MGRS) was introduced in [5]. In this approach, approximations of sets were defined through multiple equivalence relations on the universe and their properties were investigated. Using hybridization of fuzzy set [13] with rough set the concept of rough fuzzy set was introduced by Dubois and Prade [1]. Recently, a Rough Fuzzy Set Model was introduced and studied by Wu and Kou [12], which is based on Multiple Granulation. Topological properties of rough sets introduced by Pawlak in terms of their types were recently studied by Tripathy and Mitra [10]. These were extended to the context of incomplete multi granulation by Tripathy and Raghavan [11]. In this paper we introduce incomplete multigranulation on rough fuzzy sets, study their basic properties and extend the topological properties in [11] to this context. Our findings are true for both complete and incomplete fuzzy rough set models based upon multi granulation.KeywordsRough SetsFuzzy rough setsequivalence relationstolerance relationstype of rough setsmulti granular fuzzy rough sets