A Model Companion for Abelian Lattice-Ordered Groups with a Valuation
John Stokes-Waters

TL;DR
This paper develops a model companion for abelian lattice-ordered groups with valuations, using spectral subspaces and zero-set maps, achieving quantifier elimination in an expanded language.
Contribution
It introduces two multi-sorted extensions inspired by zero-set maps and proves the existence of a complete, quantifier-eliminable model companion.
Findings
The expansion is equivalent to equipping G with a spectral subspace and a specific map.
One expansion admits a complete model companion with quantifier elimination.
The results rely on a classical partial quantifier elimination theorem.
Abstract
An abelian lattice-ordered group, or abelian -group, is an abelian group equipped with a compatible lattice ordering. In this paper, we introduce two multi-sorted extensions of abelian lattice-ordered groups inspired by the zero-set maps for continuous functions into R. We demonstrate that this expansion is equivalent to equipping G with a spectral subspace X of -Spec(G), along with the map sending to . Using a classical partial quantifier elimination result originally due to Fuxing Shen and Volker Weispfenning, we show that one of these expansions admits a model companion, which is complete and has quantifier elimination in a small language expansion.
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