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Distality in Ordered Abelian Groups

Koki Okura

Abstract

We provide a characterization of distal ordered abelian groups: An ordered abelian group is distal if and only if, for each prime number $p$, the sizes of ribs with respect to the "valuation" $\mathfrak{s}_p$ are uniformly bounded. This generalizes the distality criterion for ordered abelian groups with finite spines given by Aschenbrenner, Chernikov, Gehret, and Ziegler.

Distality in Ordered Abelian Groups

Abstract

We provide a characterization of distal ordered abelian groups: An ordered abelian group is distal if and only if, for each prime number , the sizes of ribs with respect to the "valuation" are uniformly bounded. This generalizes the distality criterion for ordered abelian groups with finite spines given by Aschenbrenner, Chernikov, Gehret, and Ziegler.
Paper Structure (10 sections, 24 theorems, 10 equations)

This paper contains 10 sections, 24 theorems, 10 equations.

Table of Contents

  1. Notation and preliminaries
  2. Notation
  3. Generalities on distality
  4. Review on ordered abelian groups
  5. Valuations
  6. An example
  7. Relative quantifier elimination
  8. Proof of distality and non-distality
  9. Distality
  10. Non-distality

Key Result

Theorem 1

$G$ is distal if and only if for each prime number $p$, there exists a positive integer $N_p$ such that for every $H \in \mathcal{S}_p \setminus \left\{ \emptyset \right\}$, $|R_{H}^{\mathfrak{s}_p}| < N_p$.

Theorems & Definitions (62)

  • Theorem
  • Definition 1.1
  • Proposition 1.2: A slightly adapted version of Aschenbrenner2022DisVal
  • Proposition 1.3: cf. Aschenbrenner2022DisVal
  • Lemma 2.1
  • proof
  • Definition 2.2: Cluckers2011QuaEli
  • Lemma 2.3: Cluckers2011QuaEli
  • proof
  • Lemma 2.4
  • ...and 52 more