Transversal and Function Matroidal Structures of Covering-Based Rough Sets

In many real world applications, information blocks form a covering of a universe. Covering-based rough set theory has been proposed to deal with this type of information. It is more general and complex than classical rough set theory, hence there is much need to develop sophisticated structures to characterize covering-based rough sets. Matroids are important tools for describing graphs and linear independence of matrix theory. This paper establishes two matroidal structures of covering-based rough sets. Firstly, the transversal matroidal structure of a family of subsets of a universe is constructed. We also prove that the family of subsets of a universe is a covering if and only if the constructed transversal matroid is a normal one. Secondly, the function matroidal structure is constructed through the upper approximation number. Moreover, the relationships between the two matroidal structures are studied. Specifically, when a covering of a universe is a partition, these two matroidal structures coincide with each other.