On sticky matroids
type: Item
instance of: scholarly article
title: On sticky matroids
zbMATH Open document ID: 0639.05012
DOI: 10.1016/0012-365X(88)90173-2
author: Achim Bachem
author: Walter Kern
published in: Discrete Mathematics
publication date: 1988-01-01T00:00:00Z
review text: \textit{S. Poljak} and \textit{D. Turzik} [Discrete Math. 42, 119-123 (1982; Zbl 0491.05020)] call a geometric lattice L on the point set E sticky if for any two extensions \(L_ 1\) and \(L_ 2\) on \(E\cup X_ 1\) and \(E\cup X_ 2\), respectively, there exists a geometric lattice \(\tilde L\) on \(E\cup X_ 1\cup X_ 2\) with \(\tilde L\setminus X_ i=L_{3-i}\) for \(i=1,2\). They conjectured that a lattice is sticky if and only if it is modular, and proved this for rank 3 geometries. The present authors show ``that, in order to prove this conjecture, it is sufficient to prove it only for a quite restricted class of rank 4 geometries (which will hopefully some day turn out to be empty).'' As a simple corollary, they obtain the characterization of sticky uniform matroids [\textit{S. Poljak} and \textit{D. Turzik}, Czech. Math. J. 34(109), 239-246 (1984; Zbl 0553.05029)].
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 4041934
zbMATH Keywords: sticky geometric lattices
zbMATH Keywords: sticky uniform matroids
reviewed by: András Recski
MaRDI profile type: Publication
cites work: Q4149459
cites work: Adjoints of oriented matroids
cites work: Extension equivalence of oriented matroids
cites work: Adjoints of a Geometry
cites work: Q5619114
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description: scientific article; zbMATH DE number 4041934