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

John Stokes-Waters

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arXiv · 2603.09423

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

John Stokes-Waters

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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.

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BibTeX RIS

John Stokes-Waters. 2026-03-10. A Model Companion for Abelian Lattice-Ordered Groups with a Valuation. https://arxiv.org/abs/2603.09423

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