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.
