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T-Convexity, Tame Extensions and Definability of Hausdorff Limits in O-minimal Structures with Generic Derivations

Xiaoduo Wang

TL;DR

The work advances the model-theoretic analysis of o-minimal expansions by coupling a T-convex subring with a T-derivation. It constructs a model completion $T_{g,\text{convex}}^{\delta}$ for the combined theory and, under standard hypotheses, achieves quantifier elimination; it further proves distality and NIP in this setting. Extending to tame pairs, the paper develops a model completion $T_{g,\text{tame}}^{\delta}$, establishes stable embedding, and shows NIP, thereby generalizing CODF-type results to convex valued, tame-paired contexts. On the geometric side, it introduces definable metric topologies aligned with the $ ext{delta}$-topology and proves a Marker–Steinhorn-type definability theorem for definable types in $T^{\delta}_{g}$; it then shows that Hausdorff limits of definable families are definable, connecting model-theoretic stability to definable geometry. These results yield a cohesive framework that includes Borotta’s CODF with convex valuation as a special case and provide tools for understanding definable limits in o-minimal-differential settings.

Abstract

We study the combination of two o-minimal extensions of the theory of real closed fields: one by a T-convex subring and the other by a T-derivation. Let T be a complete, model complete o-minimal extension of RCF. We show that the combined theory T_convex^delta has a model completion T_g,convex^delta. By adding a definable unary function st, we obtain a relative quantifier elimination result for tame pairs (M, delta^M, st^M, N, delta^N, st^N), where st is the standard part map and N is Dedekind complete in M. As an application, we prove the stable embedding property for tame pairs of T_g^delta. We also associate a sequence of definable metric topologies with models of T_g^delta and prove the Marker-Steinhorn Theorem for T_g^delta. As a consequence, Hausdorff limits of definable families are definable. A special case of our framework recovers Borotta's results on CODF with convex valuation subrings and tame pairs.

T-Convexity, Tame Extensions and Definability of Hausdorff Limits in O-minimal Structures with Generic Derivations

TL;DR

The work advances the model-theoretic analysis of o-minimal expansions by coupling a T-convex subring with a T-derivation. It constructs a model completion for the combined theory and, under standard hypotheses, achieves quantifier elimination; it further proves distality and NIP in this setting. Extending to tame pairs, the paper develops a model completion , establishes stable embedding, and shows NIP, thereby generalizing CODF-type results to convex valued, tame-paired contexts. On the geometric side, it introduces definable metric topologies aligned with the -topology and proves a Marker–Steinhorn-type definability theorem for definable types in ; it then shows that Hausdorff limits of definable families are definable, connecting model-theoretic stability to definable geometry. These results yield a cohesive framework that includes Borotta’s CODF with convex valuation as a special case and provide tools for understanding definable limits in o-minimal-differential settings.

Abstract

We study the combination of two o-minimal extensions of the theory of real closed fields: one by a T-convex subring and the other by a T-derivation. Let T be a complete, model complete o-minimal extension of RCF. We show that the combined theory T_convex^delta has a model completion T_g,convex^delta. By adding a definable unary function st, we obtain a relative quantifier elimination result for tame pairs (M, delta^M, st^M, N, delta^N, st^N), where st is the standard part map and N is Dedekind complete in M. As an application, we prove the stable embedding property for tame pairs of T_g^delta. We also associate a sequence of definable metric topologies with models of T_g^delta and prove the Marker-Steinhorn Theorem for T_g^delta. As a consequence, Hausdorff limits of definable families are definable. A special case of our framework recovers Borotta's results on CODF with convex valuation subrings and tame pairs.
Paper Structure (17 sections, 48 theorems, 72 equations)

This paper contains 17 sections, 48 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $T$-Convexity and Tame Extensions
  4. Weakly O-minimality
  5. Generic Derivations on O-minimal Structures
  6. $G$-Metrics
  7. $T$-Convexity and $T$-Derivation
  8. Model Completion of $T_{\text{convex}}^{\delta}$
  9. NIP and Distality of $T^{\delta}_{g,\text{convex}}$
  10. The Theory of Tame Pairs
  11. Model Completion of $T_{\text{tame}}^{\delta}$
  12. Stable Embedding
  13. NIP of $T^{\delta}_{g,\text{tame}}$
  14. Metric Topologies on $T^{\delta}_{g}$ and Definability of Hausdorff Limits
  15. Marker-Steinhorn Theorem
  16. ...and 2 more sections

Key Result

Theorem 2.1.1

(Theorem 3.10 and Corollary 3.13 in vandendriesLewenberg1995) Suppose that $T$ has quantifier elimination and is universally axiomatizable. Then $T_{\text{convex}}$ has quantifier elimination. Without these assumptions on $T$, the theory $T_{\text{convex}}$ is complete and model complete.

Theorems & Definitions (95)

  • Theorem 2.1.1
  • Theorem 2.1.2
  • Lemma 2.1.3
  • Lemma 2.1.4
  • Remark 2.1.5
  • Lemma 2.1.6
  • Lemma 2.1.7
  • Theorem 2.2.1
  • Definition 2.2.2
  • Theorem 2.2.3
  • ...and 85 more