Matroidal structure of rough sets based on serial and transitive relations

TL;DR

This paper explores the integration of rough set theory and matroid theory by establishing a matroidal structure based on serial and transitive relations, revealing new insights into their interconnections.

Contribution

It introduces a novel matroidal framework for rough sets using serial and transitive relations and studies the inverse relation induced by matroids.

Findings

The family of minimal neighborhoods satisfies matroid circuit axioms.

Relationships between rough set approximations and matroid closure operators are established.

Connections between the two constructions of relation and matroid are analyzed.

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 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.