Residually Constructible Extensions

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 $\mathrm{dcl}$-dimension over $\mathbb{E}$ or the value group $\mathbf{val}(\mathbb{E}_*, \mathcal{O}_*)$ is $\textit{short}$ (i.e. contains no uncountable well-ordered subset). This analysis entails complete answers to [11, Problem 5.12].