A note on lattice orderability of groups
type: Item
instance of: scholarly article
title: A note on lattice orderability of groups
zbMATH Open document ID: 0564.06008
DOI: 10.1007/BF01236813
author: W. Charles Holland
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: In this elegant paper the author proves the Theorem: Lattice orderability is not a residual property; i.e., there exists a group G containing a family of normal subgroups \(\{N_ i: i\in I\}\) such that (i) \(\cap \{N_ i: i\in I\}=\{1\}\), (ii) \(G/N_ i\) is a lattice orderable group for all \(i\in I\), but (iii) G is not a lattice orderable group. The proof hinges on the obvious fact that in any lattice orderable group \(x^ 2=y^ 2\) implies \((x\vee y)x=y(x\vee y)\); so any group G which satisfies (i) \& (ii) but for which there are x \& y which are not conjugate yet \(x^ 2=y^ 2\) will be an example. The author constructs one. Note that \textit{A. A. Vinogradov} and \textit{A. A. Vinogradov} [Algebra Logika 8, 636- 639 (1969; Zbl 0264.06013)] had previously proved lattice orderability is not a local property; i.e., all finitely generated subgroups are lattice orderable but the entire group is not.
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 3898953
zbMATH Keywords: residual property
zbMATH Keywords: lattice orderable group
zbMATH Keywords: local property
reviewed by: A. M. W. Glass
MaRDI profile type: Publication
cites work: Retractable groups
cites work: An Embedding Theorem for Algebric Systems
description: scientific article; zbMATH DE number 3898953