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The domination monoid in henselian valued fields

Martin Hils, Rosario Mennuni

Abstract

We study the domination monoid in various classes of structures arising from the model theory of henselian valuations, including RV-expansions of henselian valued fields of residue characteristic 0 (and, more generally, of benign valued fields), p-adically closed fields, monotone D-henselian differential valued fields with many constants, regular ordered abelian groups, and pure short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type reductions to suitable fully embedded families of sorts in quite general settings, and full computations in concrete ones.

The domination monoid in henselian valued fields

Abstract

We study the domination monoid in various classes of structures arising from the model theory of henselian valuations, including RV-expansions of henselian valued fields of residue characteristic 0 (and, more generally, of benign valued fields), p-adically closed fields, monotone D-henselian differential valued fields with many constants, regular ordered abelian groups, and pure short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type reductions to suitable fully embedded families of sorts in quite general settings, and full computations in concrete ones.

Paper Structure

This paper contains 14 sections, 54 theorems, 12 equations.

Table of Contents

  1. Preliminaries
  2. Notation and conventions
  3. Domination
  4. A word on ⁎-types
  5. Orthogonality
  6. Regular ordered abelian groups
  7. Imaginaries in regular ordered abelian groups
  8. Moving to the right of a convex subgroup
  9. Computing the domination monoid
  10. Pure short exact sequences
  11. Finitely many definable convex subgroups
  12. Benign valued fields
  13. Mixed characteristic henselian valued fields
  14. D-henselian valued fields with many constants

Key Result

Theorem A

Let $T$ be the theory of a henselian valued field of equicharacteristic $0$, or algebraically maximal Kaplansky, possibly enriched on ${}$ and $\Gamma$. If all ${}^\times/({}^\times)^n$ are finite, then $\operatorname{\widetilde{Inv}}(\mathfrak U)\cong\operatorname{\widetilde{Inv}}({\mathop{\mathrm{

Theorems & Definitions (144)

  • Theorem A: Corollary
  • Theorem B: Theorem
  • Theorem C: Corollary
  • Theorem D: Corollary
  • Theorem E: Corollary
  • Theorem F: Theorem
  • Remark 1.2
  • proof
  • Proposition 1.3
  • proof
  • ...and 134 more