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

An abelian lattice-ordered group, or abelian $\mathrm{\ell }$-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 $\mathrm{\ell }$-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.