Closed-set lattice of regular sets based on a serial and transitive relation through matroids

TL;DR

This paper explores the relationship between lattices of regular sets derived from serial and transitive relations and matroids, demonstrating how lattice theory can enhance the understanding of rough sets in data mining.

Contribution

It introduces a method to construct a matroid from a semimodular lattice of regular sets and links lattice properties with matroid closed sets, advancing rough set analysis.

Findings

The collection of regular sets forms a semimodular lattice.

A matroid can be constructed from the lattice's height function.

A new approach to derive all matroid closed sets from the lattice.

Abstract

Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a serial and transitive relation on a universe, the collection of all the regular sets of the generalized rough set is a lattice. In this paper, we use the lattice to construct a matroid and then study relationships between the lattice and the closed-set lattice of the matroid. First, the collection of all the regular sets based on a serial and transitive relation is proved to be a semimodular lattice. Then, a matroid is constructed through the height function of the semimodular lattice. Finally, we propose an approach to obtain all the closed sets of the matroid from the semimodular lattice. Borrowing from matroids, results show that lattice theory provides an interesting view to investigate rough sets.