A lattice-theoretical characterization of oriented matroids
type: Item
instance of: scholarly article
title: A lattice-theoretical characterization of oriented matroids
zbMATH Open document ID: 0883.05033
published in: European Journal of Combinatorics
publication date: 1998-02-23T00:00:00Z
review text: To every finite matroid there is associated its lattice of flats \(L\), from which one can recover the matroid up to simplification, i.e. up to identifying parallel elements and removing loops. The lattices \(L\) which occur as lattices of flats of matroids can be characterized intrinsically as geometric lattices, that is, they are atomic and semimodular. Thus in some sense, matroids have a lattice theoretic characterization via their lattice of flats. As indicated by the title, this paper deals with the same question for oriented matroids. An oriented matroid can be specified by its set \({\mathcal O}\) of covectors, which are \(\{+,-,0\}\)-vectors, and these covectors form a lattice under the componentwise ordering induced by \(0,<,+,-\). One can ask whether the lattices arising in this way have some lattice-theoretic characterization. As the author points out, without any extra information this is probably too hard---the family of lattices of covectors of oriented matroids is very close to the family of face lattices of zonotopes, and hence characterizing these face lattices is nearly a large subcase of the very difficult Steinitz problem of characterizing face lattices of polytopes. The author's solution to this difficulty is to add an extra information. There is a natural cover-preserving order-reversing surjection \({\mathcal O} \rightarrow L\) from the poset of covectors of the oriented matroid to the lattice of flats of the underlying (simple) matroid, called the zero map. This zero map simply assigns to a covector the location of its \(0\)'s. The problem which he then solves is to characterize when a cover-preserving order-reversing surjection from a lattice \({\mathcal O}\) to a geometric lattice \(\L\) comes from an oriented matroid in this way. To this end, he introduces two technical conditions. The first, that the map \({\mathcal O} \rightarrow L\) be an antitonic folding, encompasses the properties one would expect for \({\mathcal O}\) to be the latti…
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 52B40
zbMATH DE Number: 1045837
zbMATH Keywords: lattice of flats
zbMATH Keywords: geometric lattices
zbMATH Keywords: oriented matroids
zbMATH Keywords: face lattices
zbMATH Keywords: zero map
zbMATH Keywords: symmetric antitonic foldings
author: Winfried. Hochstättler
reviewed by: Victor Reiner
MaRDI profile type: Publication
full work available at URL: eujc.1995.0120
OpenAlex ID: W1977259069
DOI: 10.1006/EUJC.1995.0120
description: scientific article; zbMATH DE number 1045837