Stably Embedded Pairs of Ordered Abelian Groups

TL;DR

This paper characterizes when ordered abelian groups are stably embedded in elementary extensions, providing a transfer principle and showing all types in a specific lexicographic product are definable.

Contribution

It offers a complete characterization for a broad class of ordered groups, including those with interpretable archimedean valuation, using a transfer principle.

Findings

All types in the lexicographic product ∏_{i∈ω} Z are definable.

Provides a transfer principle for valued groups.

Characterizes stable embedding in elementary extensions.

Abstract

We investigate when an ordered abelian group $G$ is stably embedded in a given elementary extension $H$. We focus on a large class of ordered groups which includes maximal ordered groups with interpretable archimedean valuation. We give a complete answer for groups in this class which takes the form of a transfer principle for valued groups. It follows in particular that all types in the lexicographic product $\prod_{i\in \omega} \mathbb{Z}$ are definable.