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.
Categories: math.LO