A categorical view of textural approximation spaces

Abstract

The basic motivation of the theory of textures is to find a convenient point-set based setting for fuzzy sets. Recent works on textures show that they also provide a useful model for rough set theory. In this paper, we show that i-c spaces (interior-closure texture spaces) can be regarded as a textural rough set systems on a single universe. Then we consider the approaches containing direlations and dicovers for textural rough sets and we give some basic results related to direlations and dicovers. Considering the discrete textures we discuss on these results for rough sets based on relations and coverings. Finally, we prove that the category of topological spaces and continuous functions is isomorphic to the category of (covering) approximation spaces and continuous functions. (2010): 54A40, 06A15, 18B30. Mathematics Subject Classification Introduction.Rough set theory is an extension of set theory as a mathematical approach to information systems represented by equivalence relations [15]. Generalizations of rough sets using arbitrary relations or coverings provide important real-life applications in natural computing [18][19][20][21].The starting point of the theory of textures is a point-set based setting for fuzzy sets [2,4]. Recent works on textures show that they also provide a useful model for rough set theory [8]. In this paper, we show that i-c spaces (interior-closure texture spaces) studied in [12, 13] can be regarded as a textural rough set systems on a single universe. Then we consider the approaches containing direlations and dicovers for textural rough sets and we give some basic results related to direlations and dicovers. Finally, considering the discrete textures we discuss on these results for rough sets based on relations and coverings.If interior and closure operators on a universe are grounded, isotonic, contractive (resp. expansive) and sublinear, then the universe with these operators is called an i-c space. Recall that i-c spaces (interior-closure texture spaces) and bicontinuous difunctions form a category denoted by dfIC [12]. Likewise, complemented ic spaces and bicontinuous complemented difunctions form a category denoted by cdfIC. Here, we work on i-c spaces which we call i-c * spaces whose interior and closure operators are grounded and sublinear where sublinearity holds for

Quotient structure of interior-closure texture spaces

“…Textures offer a convenient setting for the investigation of complement-free concepts in general, so much of the recent work [11][12][13][14] has proceeded independently of the fuzzy setting. In particular, the notion of interior-closure space has been introduced in [15,16] and Section 2.…”