Residually Constructible Extensions
Pietro Freni, Angus Matthews
TL;DR
This work characterizes when res-constructible extensions in o-minimal theories with T-convex valuations remain stable under taking right-factors. It defines res-constructions and proves that, for a res-constructible extension, the intermediate extensions are res-constructible exactly when either the dcl-dimension dim_dcl(E_*/E) is countable or the value group contains no uncountable ordinal. The results provide a complete answer to Tressl's problem under these conditions and connect to pseudo-completions, density, and cofinality considerations. The paper also identifies a non-local pathology showing that res-constructibility can fail to be preserved in all contexts, highlighting the delicate interaction between residue data and value group structure in T-convex valued o-minimal theories.
Abstract
Let $T$ be an o-minimal theory expanding $\mathrm{RCF}$ and $T_{\mathrm{convex}}$ be the common theory of its models expanded by predicate for a non-trivial $T$-convex valuation ring. We call an elementary extension $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_*, \mathcal{O}_*) \models T_{\mathrm{convex}}$ $\textit{res-constructible}$ if there is a tuple $\overline{s}$ in $\mathcal{O}_*$ such that $\mathbb{E}_*=\mathrm{dcl}(\mathbb{E},\overline{s})$, and the projection $\mathbf{res}(\overline{s})$ of $\overline{s}$ in the residue field sort is $\mathrm{dcl}$-independent over the residue field $\mathbf{res}(\mathbb{E}, \mathcal{O})$ of $(\mathbb{E}, \mathcal{O})$. We study factorization properties of res-constructible extensions. Our main result is that a res-constructible extension $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_*, \mathcal{O}_*)$ has the property that all $(\mathbb{E}_1, \mathcal{O}_1)$ with $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_1, \mathcal{O}_1) \prec (\mathbb{E}_*, \mathcal{O}_*)$ are res-constructible over $(\mathbb{E}, \mathcal{O})$, if and only if $\mathbb{E}_*$ has countable dimension over $\mathbb{E}$ or the value group $\mathbf{val}(\mathbb{E}_*, \mathcal{O}_*)$ contains no uncountable well-ordered subset. This analysis entails complete answers to [9, Problem 5.12].