On a theorem by Benson and Conway
type: Item
instance of: scholarly article
title: On a theorem by Benson and Conway
zbMATH Open document ID: 1105.06006
author: Jürgen Müller
published in: Journal of Pure and Applied Algebra
publication date: 2006-12-07T00:00:00Z
review text: Descriptions of a modular lattice of finite height as subspace lattice of a geometry on the ordered point set of join irreducibles have been given by \textit{U. Faigle} and \textit{C. Herrmann} [Trans. Am. Math. Soc. 266, 319--332 (1981; Zbl 0466.51001)], \textit{D. J. Benson} and \textit{J. H. Conway} [J. Pure Appl. Algebra 37, 111--116 (1985; Zbl 0575.06008)], and \textit{C. Herrmann} and \textit{M. Wild} [J. Algebra 136, 17--36 (1991; Zbl 0725.16014)]. The case of infinite height has been dealt with by \textit{C. Herrmann, D. Pickering} and \textit{M. Roddy} [Algebra Univers. 31, 365--396 (1994; Zbl 0816.06008)]. In particular, the congruence structure of the lattice and subdirect decompositions can be captured in these geometries via connected components. Including all collinearities leads to a handy axiomatization. For the purposes of representation theory and for calculations a minimum number is more convenient: it suffices to consider, for each \(a\) which is a join of two points, a single maximal set of points any two of which have join \(a\), a so-called (dotted) line. The author reviews these results and techniques and points out how they can be used in the computation of submodule lattices and their decomposition into directly indecomposables.
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 5078557
zbMATH Keywords: submodule lattices
reviewed by: Christian Herrmann
MaRDI profile type: Publication
full work available at URL: j.jpaa.2005.11.007
OpenAlex ID: W2000243628
cites work: Diagrams for modular lattices
cites work: Q5331549
cites work: Q3956998
cites work: Q3955560
cites work: Peakword condensation and submodule lattices: An application of the Meat- Axe
DOI: 10.1016/J.JPAA.2005.11.007
description: scientific article; zbMATH DE number 5078557