An example in the model theory of Abelian lattice-ordered groups
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.
existentially complete abelian lattice ordered groups
archimedean existentially complete lattice-ordered group
scientific article; zbMATH DE number 3875355