On Fuzzy Rough Set Algebras in Infinite Universes

A fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a fuzzy/crist set in a fuzzy approximation space. A fuzzy rough set algebra is a fuzzy set algebra with added dual pair of fuzzy rough approximation operators. In this paper, we study the mathematical structures of fuzzy rough set algebras in infinite universes of discourse. We first define the concept of fuzzy rough set algebras by the axiomatic approach. We then examine the properties of fuzzy rough approximation operators in different types of fuzzy rough set algebras. We also prove that if a system (F(U), boolean AND, boolean OR, similar to, L, H) is a (respectively, a serial; a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra then the derived system (F(U), boolean AND, boolean OR, similar to, LL, HH) is also a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra.