Affiliations: Department of Earth and Planetary Sciences, Graduate School of Science, Kobe University, 1-1, Rokkodaicho, Nada, Kobe, 657-8501, Japan | PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama, Japan, (PRESTO: Precursory Research for Embryonic Science and Technology) tharuna@penguin.kobe-u.ac.jp | Department of Earth and Planetary Sciences, Graduate School of Science, Kobe University, 1-1, Rokkodaicho, Nada, Kobe, 657-8501, Japan. yukio@kobe-u.ac.jp
Note: [] Address for correspondence: Department of Earth and Planetary Sciences, Graduate School of Science, Kobe University
Abstract: We explore lattice theoretic aspects in rough set theory in terms of the duality between algebra and representation. Approximation spaces are dual to complete atomic Boolean algebras in the sense that there is an adjunction between corresponding suitable categories. This is an analogy with the adjunction between the category of topological spaces and the opposite of the category of frames in pointless topology. In this paper we consider a generalization of approximation spaces called double approximation systems. A double approximation system consists of a set and two equivalence relations on it. We construct an adjunction generalizing this concept for approximation spaces. To achieve this goal, we first introduce a natural generalization of complete atomic Boolean algebras called complete prime lattices. Then we select double approximation systems that can be dual to complete prime lattices and prove the adjunction.
