Stable reducts of elementary extensions of Presburger arithmetic

TL;DR

The paper proves that stable reducts of elementary extensions of Presburger arithmetic that expand the group are essentially the same as the original group, extending known results from integers to more general abelian groups.

Contribution

It generalizes the classification of stable reducts from $(bZ, +, <)$ to elementary extensions of Presburger arithmetic, including groups like $bQ$ and $bZ$, answering a question by Conant.

Findings

Stable reducts of elementary extensions are interdefinable with the original group.

Extends previous results from integers to broader classes of abelian groups.

Provides a general framework for expansions preserving algebraic closure.

Abstract

Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < \omega$, $[G: nG]$ is finite). Then every stable reduct of $N$ which expands $(G,+)$ (equivalently every reduct that does not add new unary definable sets) is interdefinable with $(G,+)$. This extends previous results on stable reducts of $(\mathbb{Z}, +, <)$ to (stable) reducts of elementary extensions of $\mathbb{Z}$. In particular this holds for $G = \mathbb{Z}$ and $G = \mathbb{Q}$. As a result we answer a question of Conant from 2018. This result is a corollary of a more general statement about expansions of weakly-minimal 1-based expansions of abelian groups with small quotients preserving the algebraic closure operator.