Rough Algebras and 3-Valued Lukasiewicz Algebras

Description of the pairs 〈low approximation, upper approximation〉 of rough sets is an important aspect of algebraic method. By defining some basic operators, rough algebras can be constructed. Then some general algebras are 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 of which the corresponding general algebras are regular double Stone algebra, semi-simple Nelson algebra and pre-rough algebra respectively. This paper proves that all the three classic rough algebras can be made into 3-valued Lukasiewicz algebra. Thus, a uniform structure based on 3-valued Lukasiewicz algebra for the rough algebras is built. Additionally, a more direct result, i.e. all the rough sets in an approximation space can be made into a 3-valued Lukasiewicz algebra, is shown. An example is given to show how to construct the corresponding 3-valued Lukasiewicz algebra from an information system.