Residually Constructible Extensions

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].

💡 Research Summary

The paper investigates extensions of models of an o‑minimal theory T that expands the theory of real closed fields (RCF) when equipped with a non‑trivial T‑convex valuation ring O. The authors introduce the notion of a “res‑constructible” elementary extension (E,O) ≺ (E*,O*) in the common theory T₍convex₎. An extension is called res‑constructible if there exists a tuple s in O* such that E* = dcl(E,s) and the residues res(s) are dcl‑independent over the residue field res(E,O). This concept generalizes earlier pseudo‑completion constructions and captures the idea that the new field is generated by elements whose residues add independent information to the base residue field.

The central result, Theorem A, gives a precise characterization of when a res‑constructible extension remains res‑constructible after passing to any intermediate subextension. The theorem states that for a res‑constructible extension (E,O) ≺ (E*,O*), the following are equivalent: (1) the definable‑closure (dcl) dimension of E* over E is countable, or the value group v(E*,O*) contains no uncountable well‑ordered subset (i.e., it is “short”); (2) every intermediate extension (E,O) ≺ (E₁,O₁) ≺ (E*,O*) is itself res‑constructible over (E,O); (3) the same holds for all intermediate extensions that share the same residue field as (E*,O*). Thus, countable dcl‑dimension or shortness of the value group are exactly the conditions guaranteeing closure of the class of res‑constructible extensions under taking right‑factors.

The proof proceeds in two main parts. First, when the dcl‑dimension is countable, the authors build the required intermediate constructions by transfinite induction on an ordinal‑indexed sequence of generators. At successor stages the construction is straightforward; the difficulty lies at limit stages, where one must ensure that the accumulated generators still satisfy a model‑theoretic orthogonality condition between the rv‑sort and the residue sort, as well as a closure condition (Definition 25) relative to a fixed O*‑res‑construction of E* over E. Second, when the value group is short, a similar inductive scheme works because any uncountable well‑ordered chain in the value group would obstruct the necessary independence of residues.

The paper also distinguishes between power‑bounded and exponential o‑minimal theories. In the power‑bounded case, the “rv‑property” holds: every element outside the base field falls into exactly one of three mutually exclusive categories—weakly O‑immediate, O‑residual, or purely valuational. This trichotomy (Fact 12, Fact 13) simplifies the analysis of res‑constructibility, as Lemma 18 shows that elementary residue sections lift through res‑constructible extensions, and Lemma 19 proves that the isomorphism type of a res‑constructible extension is determined solely by the isomorphism type of the induced residue field extension.

Finally, the authors apply Theorem A to answer Problem 5.12 from