JournalsjemsVol. 18, No. 4pp. 681–731

Matroids over a ring

  • Alex Fink

    Queen Mary University of London, UK
  • Luca Moci

    Université Paris-Diderot Paris 7, France
Matroids over a ring cover
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Abstract

We introduce the notion of a matroid MM over a commutative ring RR, assigning to every subset of the ground set an RR-module according to some axioms. When RR is a field, we recover matroids. When R=ZR = \mathbb Z, and when RR 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 RR 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 RR. Furthermore, we compute the Tutte–Grothendieck ring of matroids over RR. We also show that the Tutte quasi-polynomial of a matroid over Z\mathbb Z can be obtained as an evaluation of the class of the matroid in the Tutte–Grothendieck ring.

Cite this article

Alex Fink, Luca Moci, Matroids over a ring. J. Eur. Math. Soc. 18 (2016), no. 4, pp. 681–731

DOI 10.4171/JEMS/600