Rough sets and matroids from a lattice-theoretic viewpoint

This paper studies rough sets via matroidal approaches from a lattice-theoretic viewpoint. We firstly give a new interpretation of definable sets of Pawlak rough set model, i.e., the set of definable sets defines uniquely a matroid, in which it is the family of open and closed sets. Then we induce two equivalence relations on a given universe based on a matroid defined on this universe. One of the equivalence relations actually is defined on the set of all atoms of a geometric lattice corresponding to the matroid, another is based on the transitivity of circuits. Properties of these two equivalence relations are then studied. Besides, we also investigate the connections between relation-based rough sets and matroids. Finally, we point out that a geometric lattice can induce a series of coverings of a universe, on which the corresponding matroid is defined, and further relations of approximations based on the induced coverings are studied.