A Relational Approach to Matroids, Simplicial Complexes and Finite Closures

TL;DR

This paper demonstrates that every finite closure operator can be represented as a matroid with bases derived from minimal covers, integrating database theory concepts into the analysis of finite closures and simplicial complexes.

Contribution

It introduces a relational approach to finite closures, showing their correspondence to matroids and simplicial complexes, and emphasizes the role of database theory in this context.

Findings

Every finite closure operator is the ground set of a matroid.

Bases of these matroids are minimal covers determining the closure.

Database theory concepts underpin the analysis of finite closures.

Abstract

The main result is Theorem MAT 11 which states that every finite closure operator is the ground set of a matroid. Its base sets consist of nonredundant covers of of the closure. These are minimal subsets that determine the closure operator using a closure algorithm from the theory of relational databases. For each hereditary collection there is one flat closure that define a matroid or simplicial complexes, but every closure defines a hereditary collection, its keys in database theory. The normalization algorithm by Maier is the basis of this paper. This follows his development, and his relevant results are cited to aid reading his original work. The main purpose of this paper is to introduce the importance of database theory into the analysis of all finite closure including the flat closures of matroids.