Roughness and fuzziness in quantales

Rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems. Motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups, rings and lattices, in this paper we introduce first the notions of fuzzy (prime, semi-prime, primary) ideals of quantales and investigate their properties. Several characterizations of such ideals are presented. Then, we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales, and establish relationships between these ideals and fuzzy prime ideals of quantales. By applying rough set theory to fuzzy ideals of quantales, we furthermore define rough fuzzy (prime, semi-prime, primary) ideals of quantales, generalizing Yang and Xu’s work on quantales to the fuzzy environment. Finally, relationships between the upper (resp. lower) rough fuzzy ideals of quantales and the upper (resp. lower) approximations of their homomorphic images are also discussed.