On the problem of dimension versus size for lattices
type: Item
instance of: scholarly article
title: On the problem of dimension versus size for lattices
zbMATH Open document ID: 0849.06007
author: Attila Sali
published in: European Journal of Combinatorics
publication date: 1996-11-04T00:00:00Z
review text: The author reduces the intriguing conjecture \(\dim (L) = o(|L|)\) for lattices to an extremal set-theoretical conjecture: \(\lim_{d \to \infty} d(\lambda(d))^{-1} = 0\), where \(d\) is the order dimension of a lattice \(L\) and \(\lambda(d)\) is the minimum size of a lattice of dimension \(d\). The paper shows a possible way of solving this conjecture by reducing it to an extremal set-theoretical one. Reduction Theorem. Let \(L\) be a lattice. Then there exists a lattice \(H\) with the following three properties: (i) \(\dim(L) \leq \dim(H) \leq \dim(L) + 1\); (ii) The set of all (meet or join) irreducible elements of \(H\) is of height 1; (iii) \(|L|\leq |H|\leq 2 \cdot |L|+ 1\). This theorem can be used to obtain lower bounds for the sizes of special lattices.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 05D05
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 895878
zbMATH Keywords: minimum size
zbMATH Keywords: extremal set-theoretical conjecture
zbMATH Keywords: order dimension of a lattice
zbMATH Keywords: irreducible elements
reviewed by: Bohumil František Šmarda
MaRDI profile type: Publication
full work available at URL: eujc.1996.0035
OpenAlex ID: W2008839164
DOI: 10.1006/EUJC.1996.0035
description: scientific article; zbMATH DE number 895878