Stable reducts of elementary extensions of Presburger arithmetic
Eran Alouf, Antongiulio Fornasiero, Itay Kaplan
TL;DR
The paper proves that for $N$ elementarily equivalent to an archimedean ordered abelian group with small quotients, every stable reduct of $N$ expanding the additive group of $G$ is interdefinable with $(G,+)$, extending Presburger-type results to elementary extensions. It introduces a preservation lemma for group homomorphisms in $0$-expansions of weakly-minimal, 1-based abelian groups with small quotients, and uses acl-equality alongside unary-definability control to deduce interdefinability. The main theorem is then specialized to archimedean ordered abelian groups, showing that stable reducts or reducts with no new unary definable sets must be interdefinable with the underlying group, with corollaries for finite rank and a positive answer to Conant’s question about definability of order on intervals. The results illuminate the rigidity of stable expansions in this setting and yield concrete consequences for when an expansion of an archimedean OAG is essentially just the original additive structure.
Abstract
Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < ω$, $[G: nG]$ 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.