Algebraic Approach to Generalized Rough Sets

Abstract

In this paper, we introduce the notion of generalized algebraic lower (upper) approximation operator and give its characterization theorem. That is, for any atomic complete Boolean algebra ${\mathcal B}$ with the set ${\mathcal A}({\mathcal B})$ of atoms, a map $L: {\mathcal B} \rightarrow {\mathcal B}$ is an algebraic lower approximation operator if and only if there exists a binary relation R on ${\mathcal A}({\mathcal B})$ such that L = R ₋, where R ₋ is the lower approximation defined by the binary relation R. This generalizes the results given by Yao.