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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.09423 |
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| _version_ | 1866913006549467136 |
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| author | Stokes-Waters, John |
| author_facet | Stokes-Waters, John |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09423 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Model Companion for Abelian Lattice-Ordered Groups with a Valuation Stokes-Waters, John Logic 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. |
| title | A Model Companion for Abelian Lattice-Ordered Groups with a Valuation |
| topic | Logic |
| url | https://arxiv.org/abs/2603.09423 |