The dimension monoid of a lattice
type: Item
instance of: scholarly article
title: The dimension monoid of a lattice
zbMATH Open document ID: 0936.06002
author: Friedrich Wehrung
published in: Algebra Universalis
publication date: 2000-02-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0501437
review text: The paper is one more step toward solving R. P. Dilworth's longstanding problem of the representation of a distributive algebraic lattice as a congruence lattice \(\text{Con }L\) for some lattice \(L\). In this work, the dimension monoid \(\text{Dim }L\) is associated with any lattice \(L\). To any pair \(a,b \in L\) an element \(\Delta (a,b)\) is given which may be viewed as a monoid-valued ``distance'' between \(a\) and \(b\) and \(\text{Dim }L\) is generated by these elements \(\Delta (a,b)\). The relations defining the \(\Delta \) function are satisfied by the mapping \(\theta \) that with any \(a,b \in L\) associates the principal congruence \(\theta (a,b)\). In particular, \(\text{Dim }L\) is a precursor of \(\text{Con }L\) and, moreover, the semilattice of all compact congruences of \(L\) is isomorphic to the maximal semilattice quotient of \(\text{Dim }L\). The dimension monoid gives much more information about \(L\) than \(\text{Con }L\) does. This dimension theory provides a generalization to all lattices of the von Neumann dimension theory for continuous geometries. In particular, if \(L\) is an irreducible continuous geometry then \(\text{Dim }L\) is either isomorphic to \(Z^+\) or to \(R^+\). If \(L\) has no infinite bounded chains then \(\text{Dim }L\) embeds into a power of \(Z^+\cup \{\infty \}\). If \(L\) is modular or \(L\) has no infinite bounded chains then \(\text{Dim }L\) is a refinement monoid. If \(R\) is a von Neumann ring and \(L\) its lattice of principal right ideals of the matrix ring \(M_2(R)\) then \(\text{Dim }L\) is isomorphic to the monoid of isomorphism classes of finitely generated projective right \(R\)-modules. Hence, the dimension theory of lattices provides a wide lattice-theoretical generalization of \(K\)-theory of regular rings. This 164 pages long paper is in fact a monograph divided into 13 chapters and including an Introduction, Contents, Index and a list of 69 references. It contains also 14 open problems concerning suc…
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06C10
zbMATH DE Number: 1407113
zbMATH Keywords: dimension monoid
zbMATH Keywords: refinement monoid
zbMATH Keywords: von Neumann dimension
zbMATH Keywords: distributive lattice
MaRDI profile type: Publication
OpenAlex ID: W2052351189
arXiv ID: math/0501437
DOI: 10.1007/S000120050091
description: scientific article