  The theory of rough sets is concerned with the lower and upper approximations
of objects through a binary relation on a universe. It has been applied to
machine learning, knowledge discovery and data mining. The theory of matroids
is a generalization of linear independence in vector spaces. It has been used
in combinatorial optimization and algorithm design. In order to take advantages
of both rough sets and matroids, in this paper we propose a matroidal structure
of rough sets based on a serial and transitive relation on a universe. We
define the family of all minimal neighborhoods of a relation on a universe, and
prove it satisfy the circuit axioms of matroids when the relation is serial and
transitive. In order to further study this matroidal structure, we investigate
the inverse of this construction: inducing a relation by a matroid. The
relationships between the upper approximation operators of rough sets based on
relations and the closure operators of matroids in the above two constructions
are studied. Moreover, we investigate the connections between the above two
constructions.
