Stable reducts of elementary extensions of Presburger arithmetic

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.