Advertisement

Russian Mathematics

, Volume 63, Issue 2, pp 18–24 | Cite as

Limitwise Monotonic Reducibility of Sets and Σ-Definability of Abelian Groups

  • D. Kh. ZainetdinovEmail author
Article

Abstract

The paper is devoted to the study of limitwise monotonic sets, as well as to the investigation of the main structural properties of limitwise monotonic reducibility (for short we will also write lm-reducibility) between sets. In this paper, we obtain a description of the algorithmic dependence between the limitwise monotonic reducibility of sets, which is defined in terms of the Σ-reducibility of the families of initial segments, and the Σ-definability of abelian groups.

Key words

limitwise monotonic function limitwise monotonic set limitwise monotonic reducibility family of subsets of natural numbers Σ-reducibility Σ-definability abelian group hereditarily finite superstructure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

4. Acknowledgments

The work was done at the expense of subsidies allocated to the Kazan Federal University for the implementation of the state assignment in the field of scientific activity (project No. 1.1515.2017/4.6).

References

  1. 1.
    Khisamiev, N. G. “Criterion for Constructivizability of a Direct Sum of Cyclic p-groups”, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 86, No. 1, 51–55 (1981).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Khoussainov, B., Nies, A., Shore, R. “Computable Models of Theories with Few Models”, Notre Dame J. Formal Logic 38, No. 2, 165–178 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kalimullin, I., Khoussainov, B., Melnikov, A. “Limitwise Monotonic Sequences and Degree Spectra of Structures”, Proc. Amer. Math. Soc. 141, No. 9, 3275–3289 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Downey, R. G., Kach, A. M., Turetsky, D. “Limitwise Monotonic Functions and their Applications”, Proceedings of the 11th Asian Logic Conf. World Sci, 59–85 (2011).Google Scholar
  5. 5.
    Zainetdinov, D. Kh., Kalimullin, I. Sh “On the Limitwise Monotonic Reducibility of Σ20-sets”, Uchen. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki 156, No. 1, 22–30 (2014).Google Scholar
  6. 6.
    Faizrahmanov, M., Kalimullin, I., Zainetdinov, D. “Maximality and Minimality under Limitwise Monotonic Reducibility”, Lobachevskii J. Math. 35, No. 4, 333–338 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kalimullin, I. Sh., Puzarenko, V. G. “Reducibility on Families”, Algebra and Logic 48, No. 1, 20–32 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zainetdinov, D. Kh. “Σ-Reducibility and lm-Reducibility of Sets and Sequences of Sets”, Uchen. Zap. Kazan. Univ. Ser. Fiz.-Mat. Nauki 158, No. 1, 51–65 (2016).MathSciNetGoogle Scholar
  9. 9.
    Ershov, Yu. L. Definability and Computability, 2nd Ed. (Ekonomika, Moscow; Nauchnaya Kniga (NII MIOONGU), Novosibirsk, 2000) [in Russian].Google Scholar
  10. 10.
    Puzarenko, V. G. “On the Computability over Models of Decidable Theories”, Algebra and Logic 39, No. 2, 98–113 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Puzarenko, V. G. “On a Certain Reducibility on Admissible Sets”, Sib. Math. J. 50, No. 2, 330–340 (2009).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia

Personalised recommendations