Relationship Between Partition Matroid and Rough Set Through k - rank Matroid ⋆

Rough set is a theory of data analysis and a mathematical tool for dealing with vagueness, incompleteness, and granularity. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. In this paper, we establish the relationships between partition matroids and rough sets through k-rank matroids. On the one hand, k-rank matroids are proposed to represent partition matroids. They reveal characteristics of partition matroids, including independence, rank function and closure operator. Through k-rank matroids, the relationship between a partition matroid and its dual matroid is studied. The circuits of partition matroids are connected with the circuits of restrictions of a k-rank matroid family. And we provide an approach to judge whether a matroid is a partition one or not. On the other hand, the lower and upper approximation operators of rough sets are expressed by some characteristics of k-rank matroids. Specially, the upper approximations are compared with the closure operator of partition matroids through k-rank matroids.