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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 429737, 16 pages
http://dx.doi.org/10.1155/2012/429737
Research Article

Matroidal Structure of Rough Sets Based on Serial and Transitive Relations

Laboratory of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, China

Received 2 August 2012; Accepted 5 November 2012

Academic Editor: Hector Pomares

Copyright © 2012 Yanfang Liu and William Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 satisfies 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.