A family of dp-minimal expansions of $(\mathbb{Z};+)$

TL;DR

This paper constructs a large family of dp-minimal structures expanding the group of integers with cyclic order, demonstrating the diversity of definable subsets within such expansions.

Contribution

It introduces a continuum of dp-minimal expansions of $( ext{Z};+)$ with distinct definable subsets, highlighting the richness of these structures.

Findings

Existence of a continuum of dp-minimal expansions

No two structures define the same subsets of $ ext{Z}$

Expansions include cyclically ordered-abelian groups

Abstract

We show that the cyclically ordered-abelian groups expanding $(\mathbb{Z};+)$ contain a continuum-size family of dp-minimal structures such that no two members define the same subsets of $\mathbb{Z}$.