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Trace Definability

Erik Walsberg

TL;DR

The paper develops a comprehensive framework for trace definability, a model‑theoretic reducibility between first‑order theories and structures. It systematically builds the theory of trace embeddings, local and $k$‑trace definability, and the $D^urg{κ}(T)$ hierarchy, proving broad equivalences and reduction principles. It then furnishes a wide array of concrete examples (covering stability, NIP, dp-rank, vector spaces, o-minimality, p-adic and real closed fields) to illustrate how theories decompose, evolve under Shelah completions, and relate to canonical bases like $ ext{RCF}$, $ ext{ACF}$, and $ ext{DLO}$. The resulting taxonomy enables transferring properties and constructing new trace equivalence classes, offering a unifying lens for comparing theories and understanding the landscape of tame vs wild behavior in model theory.

Abstract

We consider a natural notion of reducibility between first order theories.

Trace Definability

TL;DR

The paper develops a comprehensive framework for trace definability, a model‑theoretic reducibility between first‑order theories and structures. It systematically builds the theory of trace embeddings, local and ‑trace definability, and the hierarchy, proving broad equivalences and reduction principles. It then furnishes a wide array of concrete examples (covering stability, NIP, dp-rank, vector spaces, o-minimality, p-adic and real closed fields) to illustrate how theories decompose, evolve under Shelah completions, and relate to canonical bases like , , and . The resulting taxonomy enables transferring properties and constructing new trace equivalence classes, offering a unifying lens for comparing theories and understanding the landscape of tame vs wild behavior in model theory.

Abstract

We consider a natural notion of reducibility between first order theories.

Paper Structure

This paper contains 131 sections, 580 theorems, 280 equations, 1 figure.

Table of Contents

  1. Conventions and Background
  2. Homogeneous structures
  3. Airity
  4. The Shelah completion
  5. Ordinals and Cantor rank
  6. Dp-rank
  7. Op-dimension
  8. General definitions and basic facts
  9. The basics
  10. Multisorted structures and theories
  11. Infinite disjoint unions and joins
  12. Characterizations of (local) trace definability
  13. Winkler multiples and the generic variation
  14. Tarski systems and embeddings between categories of embeddings
  15. Quantifier-free trace definability between universal theories
  16. ...and 116 more sections

Key Result

Lemma 1.6

Let $\uplambda$ be a cardinal, $\mathscr{M}$ be $\uplambda$-saturated, $X \subseteq M^m$ be externally definable, and $A \subseteq M^m$ satisfy $|A| < \uplambda$. Then there is a definable $Y \subseteq M^m$ such that $X \cap A = Y \cap A$.

Figures (1)

  • Figure :

Theorems & Definitions (1059)

  • Conjecture
  • proof
  • Lemma 1.6
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • Lemma 1.11
  • proof
  • Lemma 1.12
  • proof
  • ...and 1049 more