There are no intermediate structures between the group of integers and Presburger arithmetic

TL;DR

This paper proves that any first-order structure on integers expanding addition and reduct of order plus addition must be equivalent to either just addition or addition with order, showing no intermediate structures exist.

Contribution

It establishes a rigidity result for structures on integers, demonstrating the absence of intermediate definable structures between basic additive and ordered additive structures.

Findings

No intermediate structures between $(Z,+,0)$ and $(Z,+,<,0)$ exist.

Any such structure is interdefinable with either $(Z,+,0)$ or $(Z,+,<,0)$.

The result clarifies the landscape of first-order structures on integers.

Abstract

We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or $(\mathbb{Z},+,<,0)$.