The domination monoid in henselian valued fields

Martin Hils and Rosario Mennuni

Vol. 328 (2024), No. 2, 287–323

DOI: 10.2140/pjm.2024.328.287

Abstract

We study the domination monoid in various classes of structures arising from henselian valuations, including $\mathcal{ℛ}\mathcal{𝒱}$-expansions of henselian valued fields of equicharacteristic $0$ (and, more generally, of benign valued fields), $\mathfrak{𝔭}$-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.

Keywords

model theory, invariant types, domination monoid, valued fields, ordered abelian groups

Mathematical Subject Classification

Primary: 03C45

Secondary: 03C60, 03C64, 12J10, 12L12

Milestones

Received: 10 November 2021

Revised: 13 February 2024

Accepted: 14 February 2024

Published: 30 April 2024