On Rough Algebras

Description of the pairs 〈low approximation, upper approximation〉 of rough sets is an important aspect in the research of rough set theory by algebraic method. By defining some basic operators on the approximation pairs, rough algebras can be constructed. Then some general algebras can be selected to describe the pairs of rough sets. The most famous rough algebras are Rough Double Stone Algebra, Rough Nelson Algebra and Approximation Space Algebra, and their corresponding general algebra structures are regular double Stone algebra, semi-simple Nelson algebra and pre-rough algebra respectively. This paper establishes the relations between the operators of these rough algebras and proves that: (a) approximation space algebra can be made into semi-simple Nelson algebra or regular double Stone algebra; (b) rough Nelson algebra can be made into pre-rough algebra or regular double Stone algebra; (c) rough double Stone algebra can be made into pre-rough algebra or semi-simple Nelson algebra. Thus, a uniform structure for the famous works from three different aspects is built and the relations among them are established.