# Matroids over a ring

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 4, page 681-731
- ISSN: 1435-9855

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topFink, Alex, and Moci, Luca. "Matroids over a ring." Journal of the European Mathematical Society 018.4 (2016): 681-731. <http://eudml.org/doc/277793>.

@article{Fink2016,

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.},

author = {Fink, Alex, Moci, Luca},

journal = {Journal of the European Mathematical Society},

keywords = {matroid; module over Dedekind ring; arithmetic matroid; valuated matroid; arithmetic Tutte polynomial; tropical flag Dressian; Tutte–Grothendieck ring; module over Dedekind ring; arithmetic matroid; valuated matroid; arithmetic Tutte polynomial; tropical flag Dressian; Tutte-Grothendieck ring},

language = {eng},

number = {4},

pages = {681-731},

publisher = {European Mathematical Society Publishing House},

title = {Matroids over a ring},

url = {http://eudml.org/doc/277793},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Fink, Alex

AU - Moci, Luca

TI - Matroids over a ring

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 4

SP - 681

EP - 731

AB - 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.

LA - eng

KW - matroid; module over Dedekind ring; arithmetic matroid; valuated matroid; arithmetic Tutte polynomial; tropical flag Dressian; Tutte–Grothendieck ring; module over Dedekind ring; arithmetic matroid; valuated matroid; arithmetic Tutte polynomial; tropical flag Dressian; Tutte-Grothendieck ring

UR - http://eudml.org/doc/277793

ER -

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