A matroidal approach to rough set theory

Rough set theory has been successfully applied to vague and uncertain data due to its approximation ability. Matroid is a sophisticated mathematical structure to provide a unifying abstract treatment for graph theory, linear algebra, and combinatorial optimization. In this paper, we redefine rough approximation operators through matroidal approaches and build a matroidal structure of rough set theory. First, each block of a partition is converted to a uniform matroid. In this way, a partition is transformed into a family of uniform matroids. Second, these matroids are combined through the direct sum operation to form a new matroid. Therefore the scattered uniform matroids are treated as a whole one. Third, each concept in a universe is transformed to a restriction matroid. The lower and the upper approximations of each concept are established with the matroidal approach. Fourth, for any two concepts in a universe, the relationships between the approximations of them are discussed and some new properties are revealed. These properties can be hardly found without the help of matroid theory. Fifth, the boundary region and the negative region of a concept in the universe are established directly with the matroidal approach. The lower and the upper approximations of each concept are then obtained through its boundary region. This work indicates a new approach for studying rough set theory.