Annihilator characterizations of distributivity, modularity and semimodularity
type: Item
instance of: scholarly article
title: Annihilator characterizations of distributivity, modularity and semimodularity
zbMATH Open document ID: 0930.06008
author: J. Nieminen
author: Matti Peltola
published in: Tsukuba Journal of Mathematics
publication date: 1999-09-30T00:00:00Z
full work available at URL: https://tsukuba.repo.nii.ac.jp/
review text: A graph \(G_L\) of a finite lattice \(L\) is the undirected Hasse diagram of \(L\). The distance \(d(a,b)\) between two elements \(a,b\) of \(L\) is their distance in \(G_L\). The shortest path between \(a\) and \(b\) in \(G_L\) is called an \(a\)-\(b\) geodesic. For any two elements \(a,b\) of \(L\) the geodesic annihilator \(\langle a,b\rangle_g\) is the set of all \(x\in L\) such that \(b\) is on an \(x\)-\(a\) geodesic in \(G_L\). By means of geodesic annihilators, characterizations of semimodular, modular and distributive lattices are presented. Further, conditions for a graph to be isomorphic to the graph of a semimodular, modular or distributive lattice are described.
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 05C62
zbMATH DE Number: 1309326
zbMATH Keywords: semimodularity
zbMATH Keywords: modularity
zbMATH Keywords: distributivity
zbMATH Keywords: finite lattice
zbMATH Keywords: Hasse diagram
zbMATH Keywords: geodesic annihilator
reviewed by: Bohdan Zelinka
MaRDI profile type: Publication
OpenAlex ID: W1527859345
DOI: 10.21099/TKBJM/1496163672
description: scientific article; zbMATH DE number 1309326