Geometric lattice structure of covering-based rough sets through matroids

TL;DR

This paper explores the geometric lattice structures of covering-based rough sets using matroids, establishing new relationships and conditions for approximation operators, enhancing the theoretical framework of rough set analysis.

Contribution

It constructs four geometric lattice structures of covering-based rough sets via matroids and compares their relationships, providing new theoretical insights.

Findings

Established a geometric lattice structure through transversal matroids.

Presented conditions for covering upper approximation operators to be matroid closure operators.

Compared four geometric lattice structures and analyzed core concepts like reducible elements.

Abstract

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Geometric lattice has widely used in diverse fields, especially search algorithm design which plays important role in covering reductions. In this paper, we construct four geometric lattice structures of covering-based rough sets through matroids, and compare their relationships. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by the covering, and its characteristics including atoms, modular elements and modular pairs are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, sufficient and necessary conditions for three types of covering upper approximation operators to be closure operators of matroids are presented. We exhibit three types of matroids through closure axioms, and then obtain three geometric lattice structures of covering-based rough sets. Third, these four geometric lattice structures are compared. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely geometric lattice, to study covering-based rough sets.