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Decidability and computability of certain torsion-free abelian groups. (English) Zbl 1211.03063
The paper investigates the computable properties of torsion-free abelian groups of the form $${\mathcal G}_S = \bigoplus_{n \in S} {\mathbb{Q}}_{p_n}$$, for sets $$S \subseteq \omega$$, where $$p_n$$ is the $$n$$-th prime number and $${\mathbb{Q}}_p$$ is the subgroup of $$({\mathbb{Q}}, +)$$ generated by the numbers $$1/p^k$$, for $$k \in \omega$$. It is shown that $${\mathcal G}_S$$ has a decidable copy if and only if $$S$$ is $$\Sigma_2^0$$ and has a computable copy if and only if $$S$$ is $$\Sigma_3^0$$.

MSC:
 03D45 Theory of numerations, effectively presented structures 03B25 Decidability of theories and sets of sentences 20K15 Torsion-free groups, finite rank 20K20 Torsion-free groups, infinite rank
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