On the matroidal structure of generalized rough set based on relation via definable sets

Recently, an interesting and natural research topic is to study rough set theory via matroid theory. We can introduce matroidal approaches to rough set theory and rough set methods to matroid theory which have deepened theoretical and practical significance of these two theories. In this paper, we present a systematical study on some matroidal structures of generalized rough sets based on relations. Main results are: (1) any serial relation can induce a matroid, and the upper approximation operator of the relation is not equal to the matroidal closure operator; (2) similarly, any reflexive relation can induce a matroid, and it is proved that the matroidal structure induced by any reflexive relation is equal to one induced by the symmetric (transitive, symmetric and transitive, transitive and symmetric, or equivalence) closure of the relation; (3) when a relation is reflexive, the upper approximation operator of its equivalence closure is the closure operator of the matroid induced by the relation; (4) based on the above conclusions, we prove there is a one-to-one correspondence between the upper approximation operator induced by any equivalence relation and the closure operator of any 2-circuit matroid.