Growing Spines: Ad Infinitum et Ad Infinitesimalia

TL;DR

This paper explores the structure of ordered abelian groups, proving the existence of certain extensions, and characterizes when henselian valuations are definable in the language of rings, answering open questions in valuation theory.

Contribution

It provides a first-order characterization of ordered abelian groups related to their extensions and applies this to valuation theory, resolving questions posed by Krapp, Kuhlmann, and Link.

Findings

Existence of non-trivial ordered abelian group extensions with specific properties

First-order characterization of ordered abelian groups related to extensions

Conditions under which henselian valuations are definable in the language of rings

Abstract

We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$ such that $G\preccurlyeq G\oplus H$ for some non-trivial $H$. We apply this to characterize which ordered abelian groups (respectively fields) ensure that any henselian valuation with said value group (respectively residue field) is definable in the language of rings. This answers a question of Krapp, Kuhlmann, and Link.