Department of

# Mathematics

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for events the day of Friday, December 15, 2017.

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Friday, December 15, 2017

4:00 pm in 345 Altgeld Hall,Friday, December 15, 2017

#### The groups ℤ and ℚ with predicates for being square-free

###### Minh Chieu Tran (Illinois Math)

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.