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THE ADDITIVE GROUPS OF 
$\mathbb {Z}$ AND 
$\mathbb {Q}$ WITH PREDICATES FOR BEING SQUARE-FREE
Published online by Cambridge University Press: 05 October 2020
Abstract
We consider the structures 
$(\mathbb {Z}; \mathrm {SF}^{\mathbb {Z}})$
, 
$(\mathbb {Z}; <, \mathrm {SF}^{\mathbb {Z}})$
, 
$(\mathbb {Q}; \mathrm {SF}^{\mathbb {Q}})$
, and 
$(\mathbb {Q}; <, \mathrm {SF}^{\mathbb {Q}})$
where 
$\mathbb {Z}$
is the additive group of integers, 
$\mathrm {SF}^{\mathbb {Z}}$
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 
$\mathrm {SF}^{\mathbb {Q}}$
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.
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- © Association for Symbolic Logic 2020
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