An example in the model theory of Abelian lattice-ordered groups
type: Item
instance of: scholarly article
title: An example in the model theory of Abelian lattice-ordered groups
zbMATH Open document ID: 0549.06011
DOI: 10.1007/BF01191489
author: Dan Saracino
author: Carol Wood
published in: Algebra Universalis
publication date: 1984-01-01T00:00:00Z
review text: In [Trans. Am. Math. Soc. 261, 255-270 (1980; Zbl 0482.06011)] \textit{A. M. W. Glass} and \textit{K. R. Pierce} investigated existentially complete abelian lattice ordered groups G, that is, G has the property that if \(G\subseteq H\), and H is a lattice-ordered group satisfying a sentence \(\exists\vec u\phi (\vec u,\vec g)\) where \(\vec g\in G^ n\) and \(\phi\) is a conjunction of equations and inequations, then G satisfies the same sentence. They raised the question whether an archimedean existentially complete lattice-ordered group must be hyperarchimedean (that is, every homomorphic image is archimedean). In the paper under review, the authors give a counter example. A modification of one of the examples of Glass and Pierce, it consists of real-valued functions of compact support on a certain space, extended by a constant function and a positive function with arbitrarily small positive values. A complete understanding of this paper requires a good knowledge of the theory of lattice-ordered groups, of the relevant parts of model theory, and of certain parts of the cited paper of Glass and Pierce.
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20F60
zbMATH DE Number: 3875355
zbMATH Keywords: existentially complete abelian lattice ordered groups
zbMATH Keywords: archimedean existentially complete lattice-ordered group
MaRDI profile type: Publication
cites work: Groupes et anneaux reticules
cites work: Existentially Complete Abelian Lattice-Ordered Groups
cites work: Complete theories
cites work: Finitely Generic Abelian Lattice-Ordered Groups
full work available at URL: https://doi.org/10.1007/bf01191489
OpenAlex ID: W2014720149
description: scientific article; zbMATH DE number 3875355