Matroids over a ring

TL;DR

This paper generalizes matroids to modules over a commutative ring, unifying various existing structures like matroids over fields, integers, and Dedekind domains, and explores their algebraic properties and invariants.

Contribution

It introduces the concept of matroids over a ring, extending classical matroid theory to a broader algebraic context and describing their structure and invariants.

Findings

Unified framework for matroids over rings, including fields, integers, and Dedekind domains.

Explicit description of matroids over a Dedekind domain and their properties.

Computation of the Tutte-Grothendieck ring for matroids over a ring.

Abstract

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R=$\mathbb{Z}$, and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids i.e. tropical linear spaces, respectively. More generally, whenever R is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality), and we explicitly describe the structure of the matroids over R. Furthermore, we compute the Tutte-Grothendieck ring of matroids over R. We also show that the Tutte quasi-polynomial of a matroid over $\mathbb{Z}$ can be obtained as an evaluation of the class of the matroid in the Tutte-Grothendieck ring.