New Representations of Matroids and Generalizations

TL;DR

This paper introduces new ways to represent matroids using matrices over finite semirings, expanding the classical field-based representations to boolean and superboolean semirings, and generalizes to hereditary collections.

Contribution

It extends matroid representation theory by incorporating matrices over semirings and generalizes the concept to hereditary collections, broadening the scope of matroid representations.

Findings

Matroids decomposable into field-representable matroids have boolean representations.

Any hereditary collection can be represented over the superboolean semiring.

The approach generalizes classical matroid representation to semiring-based frameworks.

Abstract

We extend the notion of matroid representations by matrices over fields and consider new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of representations is generalized naturally to include also hereditary collections. We show that a matroid that can be directly decomposed as matroids, each of which is representable over a field, has a boolean representation, and more generally that any arbitrary hereditary collection is superboolean-representable.