Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang

Abstract

This is a contribution to the classification problem for dp-minimal expansions of $(\mathbb{Z},+)$. Let $S$ be a dense cyclic group order on $(\mathbb{Z},+)$. We use results on "dense pairs" to construct uncountably many dp-minimal expansions of $(\mathbb{Z},+,S)$. These constructions are applications of the Mordell-Lang conjecture and are the first examples of "non-modular" dp-minimal expansions of $(\mathbb{Z},+)$. We canonically associate an o-minimal expansion $\mathcal{R}$ of $(\mathbb{R},+,\times )$, an $\mathcal{R}$-definable circle group $\mathbb{H}$, and a character $\mathbb{Z}\to \mathbb{H}$ to a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,S)$. We also construct a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,{\mathrm{Val}}_p)$ from the character $\mathbb{Z}\to {\mathbb{Z}}_p^{\times }$, $k\mapsto \exp (pk)$.

preliminary version, comments are welcome\

References in corpus (1)

• An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field

Cited by in corpus (1)

• Dp-minimal profinite groups and valuations on the integers