Generalized rough approximations in LΠ1/2

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call R-inner and R-outer approximations respectively (R being any finite set of rational numbers in [0, 1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic L Pi 1/2 by means of step functions from the hypercube [0, 1](k) and taking value in an arbitrary (finite) subset of [0, 1] boolean AND Q. Then, we will show that if a fuzzy set is definable as truth table of a formula of L Pi 1/2, then both its R-inner and R-outer approximation are definable as truth table of formulas of L Pi 1/2. Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set. (C) 2007 Elsevier Inc. All rights reserved.