Dp-minimal expansions of discrete ordered abelian groups

TL;DR

This paper characterizes dp-minimal expansions of discrete ordered abelian groups, showing that under certain conditions they are essentially equivalent to the standard integer structure.

Contribution

It proves that dp-minimal expansions without nontrivial definable convex subgroups are interdefinable with the standard integer structure.

Findings

Dp-minimal expansions without nontrivial definable convex subgroups are interdefinable with (Z,<,+)

Such expansions are elementarily equivalent to the standard integers

The structure (Z,<,+) is characterized as the unique minimal dp-minimal expansion under these conditions.

Abstract

If $\mathcal{Z}$ is a dp-minimal expansion of a discrete ordered abelian group $(Z,<,+)$ and $\mathcal{Z}$ does not admit a nontrivial definable convex subgroup then $\mathcal{Z}$ is interdefinable with $(Z,<,+)$ and $(Z,<,+)$ is elementarily equivalent to $(\mathbb{Z},<,+)$.