Foundations of matroids -- Part 2: Further theory, examples, and computational methods

TL;DR

This paper develops new theoretical frameworks and computational methods for understanding matroid foundations, including minimal building block lists and lattice of flats presentations, enabling calculations for various matroid classes.

Contribution

It introduces minimal building block lists for matroid foundations and new presentations in terms of minors and lattice of flats, enhancing computational approaches.

Findings

Foundations of matroids can be expressed as colimits of small minors.

Minimal lists of building blocks for 2-connected and 3-connected matroids are established.

Concrete computations of foundations for classes like whirls and projective geometries are demonstrated.

Abstract

In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the foundations of all embedded minors of M isomorphic to one of the matroids $U^2_4$, $U^2_5$, $U^3_5$, $C_5$, $C_5^\ast$, $U^2_4\oplus U^1_2$, $F_7$, $F_7^\ast$, and we show that this list is minimal. We establish similar minimal lists of building blocks for the classes of 2-connected and 3-connected matroids. We also establish a presentation for the foundation of a matroid in terms of its lattice of flats. Each of these presentations provides a useful method to compute the foundation of certain matroids, as we illustrate with a number of concrete examples. Combining these techniques with other results in the literature, we are able to compute the foundations of several interesting classes of matroids, including whirls, rank-2 uniform matroids, and projective geometries. In an appendix, we catalogue various 'small' pastures which occur as foundations of matroids, most of which were found with the assistance of a computer, and we discuss some of their interesting properties.