Condition for covering-based upper approximation operators to be closure operators of matroids

摘要

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. In order to broaden the application and theoretical areas of rough sets and matroids, some authors have combined them from many different viewpoints, such as circuits, rank function and spanning sets. In this paper, we study the relationship between five types of covering-based upper approximation operators and closure operators of matroids. On one hand, comparing those characterisations of five types of covering-based upper approximation operators with the characterisations of closure operators of matroids, we discuss the condition under which five types of covering-based upper approximation operators form closure operators of matroids and get some results. For example, the necessary and sufficient conditions for the second type of covering-based upper approximation operator to satisfy idempotence is obtained. On the other hand, by using the characterisations of unary coverings, close friends, neighbourhoods, indiscernible neighbourhoods and reduct, we present necessary and sufficient conditions for five types of covering-based upper approximation operators to be closure operators of matroids.