Matroids over a ring

Fink, 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 -