Journal of Mathematical Sciences

, Volume 211, Issue 6, pp 738–746 | Cite as

2-Computably Enumerable Degrees of Categoricity for Boolean Algebras with Distinguished Automorphisms

Article

We prove that any 2-computably enumerable Turing degree is the degree of categoricity for some computable Boolean algebra with a distinguished automorphism. Bibliography: 20 titles.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Fröhlich and J. Shepherdson, “Effective procedures in field theory,” Philos. Trans. Roy. Soc. London, Ser. A 248, 407–432 (1956).MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    A. I. Mal’tsev, “Constructive algebras I” [in Russian], Usp. Mat. Nauk 16, No. 3, 3–60 (1961).Google Scholar
  3. 3.
    S. Goncharov, V. Harizanov, J. Knight, C. McCoy, R. Miller, and R. Solomon, “Enumerations in computable structure theory,” Ann. Pure Appl. Logic 136, No. 3, 219–246 (2005).MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Chisholm, E. B. Fokina, S. S. Goncharov, V. S. Harizanov, J. F. Knight, and S. Quinn, “Intrinsic bounds on complexity and definability at limit levels,” J. Symb. Log. 74, No. 3, 1047–1060 (2009).MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    E. B. Fokina, I. Kalimullin, and R. Miller, “Degrees of categoricity of computable structures,” Arch. Math. Logic 49, No. 1, 51–67 (2010).MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    B. F. Csima, J. N. Y. Franklin, and R. A. Shore, “Degrees of categoricity and the hyperarithmetic hierarchy,” Notre Dame J. Formal Logic 54, No. 2, 215–231 (2013).MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    S. S. Goncharov, “Degrees of autostability relative to strong constructivizations” [in Russian], Tr. Mat. Inst. Steklova 274, 119–129 (2011); English transl.: Proc. Steklov Inst. Math. 274, No. 1, 105–115 (2011).Google Scholar
  8. 8.
    S. S. Goncharov and V. D. Dzgoev, “Autostability of models” [in Russian], Algebra Logika 19, No. 1, 45–58 (1980); English transl.: Algebra Logic 19, No. 1, 28–37 (1980).Google Scholar
  9. 9.
    J. B. Remmel, “Recursive isomorphism types of recursive Boolean algebras,” J. Symb. Log. 46, No. 3, 572–594 (1981).MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ch. F. D. McCoy, “Δ20 -categoricity in Boolean algebras and linear orderings,” Ann. Pure Appl. Logic 119, No. 1–3, 85–120 (2003).Google Scholar
  11. 11.
    Ch. F. D. McCoy, “Partial results in Δ30-categoricity in linear orderings and Boolean algebras” [in Russian], Algebra Logika 41, No. 5, 531–552 (2002); English transl.: Algebra Logic 41, No. 5, 295–305 (2002).Google Scholar
  12. 12.
    P. E. Alaev, “Autostable I-algebras” [in Russian], Algebra Logika 43, No. 5, 511–550 (2004); English transl.: Algebra Logic 43, No. 5, 285–306 (2004).Google Scholar
  13. 13.
    N. A. Bazhenov, ‘Δ20-Categoricity of Boolean algebras” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 13, No. 2, 3–14 (2013); English transl.: J. Math. Sci., New York 203, No. 4, 444–454 (2014).Google Scholar
  14. 14.
    V..I. Mart’yanov, “Undecidability of the theory of Boolean algebras with automorphism” [in Russian], Sib. Mat. Zh. 23, No. 3, 147–154 (1982); English transl.: Sib. Math. J. 23, No. 3, 408–415 (1983).Google Scholar
  15. 15.
    D. E. Pal’chunov and A. V. Trofimov, “Automorphisms of Boolean algebras definable by fixed elements” [in Russian], Algebra Logika 51, No. 5, 623–637 (2012); English transl.: Algebra Logic 51, No. 5, 415–424 (2012).Google Scholar
  16. 16.
    R. Tukhbatullina “Autostable of Boolean algebra Bω, enriched by automorphism” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 10, No. 3, 110–118 (2010).Google Scholar
  17. 17.
    S. S. Goncharov, Countable Boolean Algebras and Decidability [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1996); English transl.: Kluwer Academic/Plenum Publishers, New York, etc. (1997).Google Scholar
  18. 18.
    S. S. Goncharov and Yu. L. Ershov, Constructive Models [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1999); English transl.: Kluwer Academic/Plenum Press, New York (2002).Google Scholar
  19. 19.
    C. J. Ash and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Elsevier, Amsterdam (2000).MATHGoogle Scholar
  20. 20.
    N. A. Bazhenov and R. R. Tukhbatullina, “Constructivizability of the Boolean algebra B(ω) with a distinguished automorphism” [in Russian], Algebra Logika 51, No. 5, 579–607 (2012); English transl.: Algebra Logic 51, No. 5, 384–403 (2012).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations