A cover-preserving embedding of semimodular lattices into geometric lattices
type: Item
instance of: scholarly article
title: A cover-preserving embedding of semimodular lattices into geometric lattices
zbMATH Open document ID: 1226.06004
author: Gábor Czédli
author: E. T. Schmidt
published in: Advances in Mathematics
publication date: 2010-10-07T00:00:00Z
review text: The authors prove that for any semimodular lattice \(\mathcal L\) of finite length there exists a cover-preserving embedding into a geometric lattice \(G({\mathcal L})\) of the same length such that the number of atoms of \(G({\mathcal L})\) coincides with the number of join-irreducible elements of \(\mathcal L\). For finite semimodular lattices this has been shown by \textit{G. Grätzer} and \textit{E. W. Kiss} [Order 2, 351--365 (1986; Zbl 0586.06001)], and again by \textit{M. Wild} [Discrete Math. 112, No.~1--3, 207--244 (1993; Zbl 0808.06009)] with a short proof using matroid theory. It is pointed out that the finite geometric lattice constructed by Grätzer and Kiss differs from \(G({\mathcal L})\). Detailed historical notes are given, see also Chapter 6.2 in [\textit{M. Stern}, Semimodular lattices. Theory and applications. Cambridge: Cambridge University Press (1999; Zbl 0957.06008)].
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 5796918
zbMATH Keywords: semimodular lattice
zbMATH Keywords: geometric lattice
zbMATH Keywords: cover-preserving embedding
zbMATH Keywords: matroid
reviewed by: Horst Szambien
MaRDI profile type: Publication
full work available at URL: j.aim.2010.05.001
OpenAlex ID: W1976424510
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DOI: 10.1016/J.AIM.2010.05.001
description: scientific article; zbMATH DE number 5796918