Weak maps and stabilizers of classes of matroids
type: Item
instance of: scholarly article
title: Weak maps and stabilizers of classes of matroids
zbMATH Open document ID: 0919.05010
author: James Oxley
author: Dirk Vertigan
author: James F. Geelen
author: Geoff Whittle
published in: Advances in Applied Mathematics
publication date: 1999-04-06T00:00:00Z
review text: Let \(N\) be a matroid belonging to a class \({\mathcal N}\) of matroids representable over a field \(F\), and assume that \({\mathcal N}\) is closed under minors and duals. \(N\) is an \(F\)-stabilizer of \({\mathcal N}\) if every representation of any 3-connected member of \({\mathcal N}\) is determined (up to elementary row and column operations) by any \(N\)-minor. A universal stabilizer is an \(F\)-stabilizer for every \(F\) over which the class \({\mathcal N}\) is respresentable. The authors examine the basic properties of universal stabilizers and prove the following theorem. (Note that an element is fixed (cofixed) in \(M\) if there is no elementary extension (coextension) of \(M\) by an element \(x'\) which is equivalent to \(x\) but not dependent on \(x\).) Theorem 6.1: Let \(N\) be a 3-connected matroid in a well-closed class \({\mathcal N}\) and suppose that \(| E(N)| \geq 2\). Then \(N\) is a universal stabilizer for \({\mathcal N}\) if and only if the following three conditions hold: (i) If \(M\) is a 3-connected member of \({\mathcal N}\) with an element \(x\) such that \(M \setminus x = N\), then \(x\) is fixed in \(M\). (ii) If \(M\) is a 3-connected member of \({\mathcal N}\) with an element \(y\) such that \(M/y = N\), then \(y\) is cofixed in \(M\). (iii) If \(M\) is a 3-connected member of \({\mathcal N}\) with a pair of elements \(x\) and \(y\) such that \(M \setminus x / y = N\) and \(M \setminus x\) is 3-connected, then \(x\) is fixed in \(M\).
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1221598
zbMATH Keywords: weak map
zbMATH Keywords: representable matroid
zbMATH Keywords: quaternary matroid
reviewed by: Herman J. Servatius
MaRDI profile type: Publication
OpenAlex ID: W1975424019
cites work: On Reid's characterization of the ternary matroids
cites work: A simple theorem on 3-connectivity
cites work: On relative position in extensions of combinatorial geometries
cites work: Single-element extensions of matroids
cites work: Unavoidable minors of large 3-connected matroids
cites work: Q3816080
cites work: The excluded minors for GF(4)-representable matroids
cites work: On the Uniqueness of Matroid Representations Over GF(4)
cites work: Weak Maps of Combinatorial Geometries
cites work: Q3135082
cites work: On inequivalent representations of matroids over finite fields
cites work: On weak maps of ternary matroids
cites work: A note on the non-spanning circuits of a matroid
cites work: Partial fields and matroid representation
cites work: Matroid representation over GF(3)
cites work: Decomposition of regular matroids
cites work: Q3249424
cites work: Connectivity in Matroids
cites work: Latroids and their representation by codes over modules
cites work: A characterisation of the matroids representable over GF(3) and the rationals
cites work: On Matroids Representable over 𝐺𝐹(3) and Other Fields
cites work: Stabilizers of classes of representable matroids
DOI: 10.1006/AAMA.1998.0600
description: scientific article; zbMATH DE number 1221598