Applications of Matrices to a Matroidal Structure of Rough Sets

Rough sets provide an efficient tool for dealing with the vagueness and granularity in information systems. They are widely used in attribute reduction in data mining. There are many optimization issues in attribute reduction. Matroids generalize the linear independence in vector spaces and are widely used in optimization. Therefore, it is necessary to integrate rough sets and matroids. In this paper, we apply matrices to a matroidal structure of rough sets through three sides, which are characteristics, operations, and axioms. Firstly, a matroid is induced by an equivalence relation, and the matroid is a representable matroid whose representable matrix is a matrix representation of the equivalence relation. Then some characteristics of the matroid are presented through the representable matrix mainly. Secondly, contraction and restriction operations are applied to the matroid through the representable matrix and approximation operators of rough sets. Finally, two axioms of circuit incidence matrices of 2-circuit matroids are obtained, where 2-circuit matroids are proposed based on the characteristics of the matroid. In a word, these results show an interesting view to investigate the combination between rough sets and matroids through matrices.