On the number of join irreducibles and acyclicity in finite modular lattices
type: Item
instance of: scholarly article
title: On the number of join irreducibles and acyclicity in finite modular lattices
zbMATH Open document ID: 1216.06004
author: Christian Herrmann
published in: Algebra Universalis
publication date: 2011-05-11T00:00:00Z
review text: Motivated by the relations between modular lattices and geometry, the author introduces the notion of acyclicity for subsets of a finite modular lattice \(L\). It turns out that \(L\) is acyclic if it admits some acyclic base of lines. Let \(d(L)\) and \(s(L)\) be the height of \(L\) and the number of maximal congruences of \(L\), respectively. The main result of the paper is as follows. Some (equivalently: any) base of \(L\) is acyclic if and only if \(L\) has exactly \(2d(L)-s(L)\) join irreducibles.
reviewed by: Ján Jakubík
Mathematics Subject Classification ID: 06C05
zbMATH DE Number: 5886745
zbMATH Keywords: modular lattice
zbMATH Keywords: acyclicity
zbMATH Keywords: join irreducibility
MaRDI profile type: Publication
full work available at URL: s00012-011-0112-9
OpenAlex ID: W2079608047
cites work: Diagrams for modular lattices
cites work: Q3956998
cites work: Projective Geometry on Partially Ordered Sets
cites work: Acyclic modular lattices and their representations
cites work: A geometric description of modular lattices
cites work: On locally finite modular lattice varieties of finite height
cites work: Q5686073
cites work: Q3770580
cites work: The minimal number of join irreducibles of a finite modular lattice
cites work: Subdirekte Produkte vollständiger Verbände.
DOI: 10.1007/S00012-011-0112-9
description: scientific article; zbMATH DE number 5886745