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

John Stokes-Waters

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.

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

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.
Paper Structure (16 sections, 32 theorems, 67 equations)

This paper contains 16 sections, 32 theorems, 67 equations.

Table of Contents

  1. Discussion
  2. Preliminaries
  3. Valuations on $\ell$-Groups
  4. Motivation
  5. The Natural Valuation of an $\ell$-Group
  6. Topological Representations of Valued $\ell$-Groups
  7. Conrad-Darnel-Nelson Valuations
  8. Standard Structures
  9. Functional Representations of Densely Valued $\ell$-Groups
  10. The Shen-Weispfenning Theorem
  11. Algebraically Closed Densely Valued $\ell$-Groups
  12. Existentially Closed Densely Valued $\ell$-Groups
  13. The Model Theory of Existentially Closed Densely Valued $\ell$-Groups
  14. Completeness
  15. Quantifier Elimination
  16. ...and 1 more sections

Key Result

Theorem 3.2

Let $\mathord{\mathcal{G}}$ be an $\ell$-group. Then, there are $\ell$-group embeddings: for some divisible $o$-group $\Gamma$. We remark that this map isn't uniquely determined, but for any choice of composition $I$, we have that: for any $f \in \mathord{\mathcal{G}}$ and $\mathord{\mathcal{J}} \in \ell\textnormal{-Spec}(\mathord{\mathcal{G}})$.

Theorems & Definitions (76)

  • Example 3.1
  • Theorem 3.2
  • Definition 3.3: Valuation on an $\ell$-Group
  • Example 3.4
  • Definition 3.5: Valued $\ell$-Group Morphism
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • Corollary 3.8: The Natural Valuation Adjunction
  • ...and 66 more