The domination monoid in henselian valued fields

TL;DR

This paper investigates the structure of the domination monoid across various classes of henselian valued fields and related structures, providing reductions and explicit computations in broad and specific contexts.

Contribution

It introduces a unified approach to analyze the domination monoid in henselian valued fields and related structures, with new reductions and explicit calculations.

Findings

Ax-Kochen-Ershov type reductions established

Full computations of the domination monoid in concrete cases

Unified framework for various classes of 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.