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 $(\mathbb{E},\mathcal{O})\prec ({\mathbb{E}}_{\ast },{\mathcal{O}}_{\ast })\models T_{\mathrm{convex}}$ $\text{res-constructible}$ if there is a tuple $\overline{s}$ in ${\mathcal{O}}_{\ast }$ such that ${\mathbb{E}}_{\ast }=\mathrm{dcl}(\mathbb{E},\overline{s})$, 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 $(\mathbb{E},\mathcal{O})\prec ({\mathbb{E}}_{\ast },{\mathcal{O}}_{\ast })$ has the property that all $({\mathbb{E}}_1,{\mathcal{O}}_1)$ with $(\mathbb{E},\mathcal{O})\prec ({\mathbb{E}}_1,{\mathcal{O}}_1)\prec ({\mathbb{E}}_{\ast },{\mathcal{O}}_{\ast })$ are res-constructible over $(\mathbb{E},\mathcal{O})$, if and only if ${\mathbb{E}}_{\ast }$ has countable $\mathrm{dcl}$-dimension over $\mathbb{E}$ or the value group $\mathbf{val}({\mathbb{E}}_{\ast },{\mathcal{O}}_{\ast })$ is $\text{short}$ (i.e. contains no uncountable well-ordered subset). This analysis entails complete answers to [11, Problem 5.12].

25 pages, Secondary classes 12J10, 12J15