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.