Enumerations and completely decomposable torsion-free abelian groups. (English) Zbl 1204.03039

Melnikov, Alexander G.

Theory Comput. Syst. 45, No. 4, 897-916 (2009).

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.

Reviewer: Simion Sorin Breaz (Cluj-Napoca)

MSC:

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

Keywords:

computable models; torsion-free abelian groups; degree spectra

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