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Enumerations and completely decomposable torsion-free abelian groups. (English) Zbl 1204.03039
The main results of this paper are the following:
For any family \(R\) of finite sets there exists a completely decomposable torsion-free abelian group \(G_{R}\) of infinite rank such that \(G_{R}\) has an \(X\)-computable copy if and only if \(R\) has a \(\Sigma_{2}^{X}\)-computable enumeration (Theorem 4).
There exists a completely decomposable torsion-free abelian group \(G\) of infinite rank such that \(G\) has an \(X\)-decomposable copy if and only if \(X^{\prime}>_{T}0^{\prime}\) (Theorem 5).
As a consequence of these results, it is proved that there exists a completely decomposable torsion-free abelian group \(G\) of infinite rank with no jump degree.

MSC:
03C57 Computable structure theory, computable model theory
03D45 Theory of numerations, effectively presented structures
20K20 Torsion-free groups, infinite rank
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