The Matroidal Structures of the Second Type of Covering-Based Rough Set

摘要

Rough set theory is a useful tool for data mining. In recent yeas, ones have combined it with matroid theory to construct an excellent set-theoretical framework for empirical machine learning methods. Hence, the study of its matroidal structure is an interesting research topic, and the structure is part of the foundation of rough set theory. Few people study the combinations the second type of covering-based rough sets with matroids. In this paper, we mainly build the matroidal structures of the second type of covering-based rough sets from the perspective of closure operators. On the one hand, we establish a closure system through the fixed point family of the second type of covering lower approximation operator, and then construct a corresponding closure operator. For a covering of a universe, this closure operator is a matroidal closure operator if and only if the reduct of the covering forms a partition of the universe. On the other hand, we present two sufficient and necessary conditions for the second type of covering upper approximation operator to form a matroidal closure operator through the indiscernible neighborhood and the covering upper approximation operator.