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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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References:
 [1] Baer, R.: Abelian groups without elements of finite order. Duke Math. J. 3, 68–122 (1937) · Zbl 0016.20303 · doi:10.1215/S0012-7094-37-00308-9 [2] Coles, R., Downey, R., Slaman, T.: Every set has a least jump enumeration. J. Lond. Math. Soc. (2) 62(3), 641–649 · Zbl 1023.03036 [3] Downey, R.: On presentations of algebraic structures. In: Sorbi (ed.) Complexity, Logic and Recursion Theory, pp. 157–205. Marcel Dekker, New York (1997) · Zbl 0915.03039 [4] Ershov, Y., Goncharov, S.: Constructive Models. Nauchnaya kniga, Novosibirsk (1999) [5] Fuchs, L.: Infinite Abelian Groups, vol. II. Academic Press, New York (1973) · Zbl 0257.20035 [6] Khisamiev, N.: Constructive Abelian Groups. Handbook of Recursive Mathematics, vol. 139. Elsevier, Amsterdam (1998) · Zbl 0940.03044 [7] Mal’tsev, A.I.: On recursive Abelian groups. Soviet Math. Dokl. 3 [8] Miller, R.: The $$\Delta$$ 2 0 -spectrum of a linear order. J. Symb. Log. 66, 470–486 (2001) · Zbl 0992.03050 · doi:10.2307/2695025 [9] Selman, A.: Arithmetical reducibilities I. Z. Math. Log. Grundlagen Math. 17, 335–370 (1971) · Zbl 0229.02037 · doi:10.1002/malq.19710170139 [10] Slaman, T.: Relative to any non-recursive set. Proc. Am. Math. Soc. 126, 2117–2122 (1998) · Zbl 0894.03017 · doi:10.1090/S0002-9939-98-04307-X [11] Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin/Heidelberg (1987) · Zbl 0667.03030 [12] Wehner, S.: Enumerations, countable structures and Turing degrees. Proc. Am. Math. Soc. 126, 2131–2139 (1998) · Zbl 0906.03044 · doi:10.1090/S0002-9939-98-04314-7
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