Excess of a lattice
type: Item
instance of: scholarly article
title: Excess of a lattice
zbMATH Open document ID: 1023.06002
author: Tetsuya Abe
published in: Graphs and Combinatorics
publication date: 2003-03-25T00:00:00Z
review text: A pair \((x,y)\) of elements in a lattice is said to be mismatching if \(x\) is join-irreducible, \(y\) is meet-irreducible and \(x\nleqslant y\). The excess of a lattice \(L\) is the number \(\text{ex}(L) =|L|-\min \{|V_x|+ |I_y|\): \((x,y)\) is a mismatching pair\}, where \(V_x\) and \(I_y\) are the filter generated by \(x\) and the ideal generated by \(y\), respectively. It is shown that a lattice has excess zero (one) iff it is isomorphic to a Boolean lattice (of one of the following types: two-element chain, diamond, pentagon). It is also proved that the cardinality of a relatively complemented lattice which is not Boolean is at most five times its excess.
reviewed by: Václav Slaví k
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B10
zbMATH DE Number: 1887502
zbMATH Keywords: excess
zbMATH Keywords: mismatching pair
zbMATH Keywords: Boolean lattice
zbMATH Keywords: diamond
zbMATH Keywords: pentagon
zbMATH Keywords: relatively complemented lattice
zbMATH Keywords: product
MaRDI profile type: Publication
full work available at URL: s003730200029
OpenAlex ID: W1963534702
DOI: 10.1007/S003730200029
description: scientific article; zbMATH DE number 1887502