Extension equivalence of oriented matroids
type: Item
instance of: scholarly article
title: Extension equivalence of oriented matroids
zbMATH Open document ID: 0629.05026
DOI: 10.1016/S0195-6698(86)80020-8
author: Achim Bachem
author: Walter Kern
published in: European Journal of Combinatorics
publication date: 1986-01-01T00:00:00Z
review text: By using a one-to-one correspondence between the point extensions of a matroid M and the set of all linear subclasses of M which, ordered by inclusion, form the so-called extension lattice of M, the authors introduce and investigate a special relation between matroids called extension equivalence. More precisely, they define two matroids to be extension equivalent if their extension lattices are isomorphic, and prove the following main results. First, they show that if M and M' are extension equivalent matroids having equal rank, then every adjoint of M is also an adjoint of M'. Second, using this result they prove that under the same assumption, M is representable over some field F if and only if M' is. And finally, the problem how to translate both the results to oriented matroids is discussed in the final section of the paper, and it is proved that they remain valid if the terms ``matroid'' and ``adjoint'' are replaced by the terms ``oriented matroid'' and ``oriented adjoint'', respectively.
reviewed by: J.Libicher
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4023307
zbMATH Keywords: point extensions
zbMATH Keywords: matroid
zbMATH Keywords: extension lattice
zbMATH Keywords: extension equivalence
zbMATH Keywords: oriented matroids
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4023307