Roughness in quantales

Rough set theory is a useful mathematical tool for dealing with the uncertainty and granularity in information systems. In this paper, rough approximations are introduced into quantales, a kind of partially ordered algebraic structure with an associative binary multiplication. The notions of (upper, lower) rough (prime, semi-prime, primary) ideals of quantales are proposed and verified to be the extended notions of usual (prime, semi-prime, primary) ideals of quantales respectively. Global order properties of all lower (upper) rough ideals of a quantale are also investigated. The rough radical of an upper rough ideal in a quantale is defined and the prime radical theorem of rings is generalized to upper rough ideals of quantales. Some results about homomorphic images of rough ideals and rough prime ideals of semigroups and rings are also generalized and improved in the field of quantales.