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The theory of rough sets is concerned with the lower and upper approximationsof objects through a binary relation on a universe. It has been applied tomachine learning, knowledge discovery and data mining. The theory of matroidsis a generalization of linear independence in vector spaces. It has been usedin combinatorial optimization and algorithm design. In order to take advantagesof both rough sets and matroids, in this paper we propose a matroidal structureof rough sets based on a serial and transitive relation on a universe. Wedefine the family of all minimal neighborhoods of a relation on a universe, andprove it satisfy the circuit axioms of matroids when the relation is serial andtransitive. In order to further study this matroidal structure, we investigatethe inverse of this construction: inducing a relation by a matroid. Therelationships between the upper approximation operators of rough sets based onrelations and the closure operators of matroids in the above two constructionsare studied. Moreover, we investigate the connections between the above twoconstructions.

Matroidal structure of rough sets based on serial and transitive relations