Skip to main content
SpringerLink
Log in

On Σ-subsets of naturals over abelian groups

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain conditions for the Σ-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in \(\mathbb{H}\mathbb{F}(A)\) coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in \(\mathbb{H}\mathbb{F}(A)\) coincides with I.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Morozov A. S. and Puzarenko V. G., “Σ-subsets of naturals,” Algebra i Logika, 43, No. 3, 291–320 (2004).

    MathSciNet  Google Scholar 

  2. Rudnev V. A., “Existence of an inseparable pair in the recursive theory of admissible sets,” Algebra i Logika, 27, No. 1, 48–56 (1988).

    MATH  MathSciNet  Google Scholar 

  3. Puzarenko V. G., “Computability over models of decidable theories,” Algebra i Logika, 39, No. 2, 170–197 (2000).

    MATH  MathSciNet  Google Scholar 

  4. Morozov A. S., “A Σ-set of natural numbers not enumerable by natural numbers,” Siberian Math. J., 41, No. 6, 1162–1166 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  5. Khisamiev A. N., “On the Ershov upper semilattice \(\mathfrak{L}_E \),” Siberian Math. J., 45, No. 1, 173–187 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  6. Barwise J., Admissible Sets and Structures, Springer-Verlag, Berlin (1975).

    Google Scholar 

  7. Ershov Yu. L., Definability and Computability [in Russian], Nauchnaya Kniga, Novosibirsk (1996).

    Google Scholar 

  8. Rogers H., Theory of Recursive Functions and Effective Computability [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  9. Kargapolov M. I. and Merzlyakov Yu. I., Fundamentals of the Theory of Groups [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  10. Kalimullin I. Sh. and Puzarenko V. G., “The computable principles on admissible sets,” Mat. Trudy, 7, No. 2, 35–71 (2004).

    MathSciNet  Google Scholar 

  11. Rudnev V. A., “A universal recursive function on admissible sets,” Algebra i Logika, 25, No. 4, 425–435 (1986).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. N. Khisamiev

Authors
  1. A. N. Khisamiev
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text Copyright © 2006 Khisamiev A. N.

The author was supported by the President of the Russian Federation (Grant MK1807.2005.1), the Russian Foundation for Basic Research (Grant 05-01-00819), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2112.2003.1), and the Program “Universities of Russia” (Grant UR.04.01.019).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 3, pp. 695–706, May–June, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khisamiev, A.N. On Σ-subsets of naturals over abelian groups. Sib Math J 47, 574–583 (2006). https://doi.org/10.1007/s11202-006-0068-8

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-006-0068-8

Keywords

Search

Navigation