Some characteristics of matroids through rough sets

TL;DR

This paper explores how rough set theory can be used to analyze and characterize matroids, establishing a new connection between these mathematical structures and deriving properties like independence and bases.

Contribution

It introduces a novel method of constructing matroids from rough set upper approximations and investigates their characteristics, bridging rough sets and matroid theory.

Findings

A family of sets from rough set upper approximations satisfies matroid support axioms.

A new class of matroids, called support matroids, is induced by equivalence relations.

Characteristics such as independent sets, bases, and hyperplanes are analyzed using rough sets.

Abstract

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to matroids through defining a family of sets which are constructed from the upper approximation operator with respect to an equivalence relation. First, we prove the family of sets satisfies the support set axioms of matroids, and then we obtain a matroid. We say the matroids induced by the equivalence relation and a type of matroid, namely support matroid, is induced. Second, through rough sets, some characteristics of matroids such as independent sets, support sets, bases, hyperplanes and closed sets are investigated.