Matrix approaches to rough sets through vector matroids over fields

Abstract

Rough sets were proposed to deal with the vagueness and incompleteness of knowledge in information systems. There are many optimisation issues in this field such as attribute reduction. Matroids generalised from matrices are widely used in optimisation issues. Therefore, it is necessary to connect matroids with rough sets. In this paper, we take fields into consideration and introduce matrices to study rough sets through vector matroids. First, a matrix representing of an equivalence relation is proposed, and then a matroidal structure of rough sets over a field is presented by the matrix. Second, the circuits of the matroidal structure are studied through matrix null spaces. Third, over a binary field, we construct an equivalence relation from a matrix null space, and find that a family of equivalence relations and a family of sets, which any member is a collection of the minimal non-empty sets that are supports of members of null space of a binary dependence matrix, are isomorphic. In a word, matrices provide an interesting viewpoint to study rough sets.