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Friday, December 15, 2017

**Abstract:** (Joint work with Neer Bhardwaj) We study the model-theoretic properties of the 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 $\mathrm{Sqf}^\mathbb{Z}$ is the set of $a \in \mathbb{Z}$ such that $v_{p}(a) < 2$ for every prime $p$ and corresponding $p$-adic valuation $v_{p}$ and $\mathrm{Sqf}^\mathbb{Q}$ is defined likewise for rational numbers. The first two structures are analogues of structures considered by Bateman, Jockusch, Woods, Kaplan, and Shelah concerning predicates for being prime. Avoiding their use of Dickson's conjecture, we obtain the first unconditional natural example of a simple unstable expansion of $(\mathbb{Z};+)$. This work can be viewed as part of the ongoing program where we investigate model-theoretic properties of structures whose components interact in a random fashion with one another.