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

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 $\ell$-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 $\ell$-Spec(G), along with the map sending $a \in G$ to $V(a \wedge 0) \cap X$. 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.