The additive groups of $\mathbb{Z}$ and $\mathbb{Q}$ with predicates for being square-free

Abstract

We consider the four structures $(\mathbb{Z};{\mathrm{Sqf}}^{\mathbb{Z}})$, $(\mathbb{Z};<,{\mathrm{Sqf}}^{\mathbb{Z}})$, $(\mathbb{Q};{\mathrm{Sqf}}^{\mathbb{Q}})$, and $(\mathbb{Q};<,{\mathrm{Sqf}}^{\mathbb{Q}})$ where $\mathbb{Z}$ is the additive group of integers, ${\mathrm{Sqf}}^{\mathbb{Z}}$ is the set of $a\in \mathbb{Z}$ such that $v_p(a)<2$ for every prime $p$ and corresponding $p$-adic valuation $v_p$, $\mathbb{Q}$ and ${\mathrm{Sqf}}^{\mathbb{Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

25 pages, published version