Matroids over a ring
Alex Fink
Queen Mary University of London, UKLuca Moci
Université Paris-Diderot Paris 7, France
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Abstract
We introduce the notion of a matroid over a commutative ring , assigning to every subset of the ground set an -module according to some axioms. When is a field, we recover matroids. When , and when 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 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 . Furthermore, we compute the Tutte–Grothendieck ring of matroids over . We also show that the Tutte quasi-polynomial of a matroid over 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