Abstract rough approximation spaces (bridging the gap between fuzziness and roughness)

Roughness as vague concepts described by two well specified ones is introduced. From a formal point of view an approximation space is a partial ordered set of approximable elements, equipped with a subset of open definable elements and a subset of closed definable elements (in general different between them); an inner approximation map and an outer approximation map furnish the best approximation from the bottom and from the top of any element. It is shown as a structure of this kind can be induced from any quasi Brouwer-Zadeh (BZ) poset and that usual approaches to generalized rough set theory (as discernibility space) and to fuzzy set theory can be described in this context