Matrix approach to rough sets through vector matroids over a field

Rough sets were proposed to deal with the vagueness and incompleteness of knowledge in information systems. There are may optimization issues in this field such as attribute reduction. Matroids generalized from matrices are widely used in optimization. Therefore, it is necessary to connect matroids with rough sets. In this paper, we take field into consideration and introduce matrix to study rough sets through vector matroids. First, a matrix representation of an equivalence relation is proposed, and then a matroidal structure of rough sets over a field is presented by the matrix. Second, the properties of the matroidal structure including circuits, bases and so on are studied through two special matrix solution spaces, especially null space. Third, over a binary field, we construct an equivalence relation from matrix null space, and establish an algebra isomorphism from the collection of equivalence relations to the collection of sets, which any member is a family of the minimal non-empty sets that are supports of members of null space of a binary dependence matrix. In a word, matrix provides a new viewpoint to study rough sets.