Advertisement

Algebra and Logic

, Volume 52, Issue 2, pp 89–97 | Cite as

Computable categoricity of the Boolean algebra \( \mathfrak{B}\left( \omega \right) \) with a distinguished automorphism

  • N. A. Bazhenov
  • R. R. Tukhbatullina
Article

It is proved that every computably enumerable Turing degree is a degree of categoricity of some computable Boolean algebra with a distinguished automorphism. We construct an example of a computably categorical Boolean algebra with a distinguished automorphism, having a set of atoms in a given computably enumerable Turing degree.

Keywords

Boolean algebra with distinguished automorphism computable categoricity categoricity spectrum degree of categoricity 

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).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    A. I. Mal’tsev, “Constructive models. I,” Usp. Mat. Nauk, 16, No. 3, 3–60 (1961).Google Scholar
  3. 3.
    S. S. Goncharov, V. S. Harizanov, J. F. Knight, C. McCoy, R. Miller, and R. Solomon, “Enumerations in computable structure theory,” Ann. Pure Appl. Log., 136, No. 3, 219–246 (2005).MathSciNetzbMATHCrossRefGoogle 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).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    E. B. Fokina, I. Kalimullin, and R. Miller, “Degrees of categoricity of computable structures,” Arch. Math. Log., 49, No. 1, 51–67 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    B. F. Csima, J. N. Franklin, and R. A. Shore, “Degrees of categoricity and the hyperarithmetic hierarchy,” Notre Dame J. Formal Log., 54, No. 2, 215–231 (2013).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    S. S. Goncharov and V. D. Dzgoev, “Autostability of models,” Algebra Logika, 19, No. 1, 45–58 (1980).MathSciNetGoogle Scholar
  8. 8.
    J. B. Remmel, “Recursive isomorphism types of recursive Boolean algebras,” J. Symb. Log., 46, No. 3, 572–594 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    C. McCoy, “Δ0 2-categoricity in Boolean algebras and linear orderings,” Ann. Pure Appl. Log., 119, Nos. 1-3, 85–120 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ch. F. McCoy, “Partial results in Δ0 3-categoricity in linear orderings and Boolean algebras,” Algebra Logika, 41, No. 5, 531–552 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    K. Harris, “Categoricity in Boolean algebras,” to appear.Google Scholar
  12. 12.
    P. E. Alaev, “Autostable I-algebras,” Algebra Logika, 43, No. 5, 511–550 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    N. A. Bazhenov, “Δ0 2-categoricity of Boolean algebras,” to appear in Vestnik NGU, Mat., Mekh., Inf. Google Scholar
  14. 14.
    R. R. Tukhbatullina, “Autostability of the Boolean algebra \( {{\mathfrak{B}}_{\omega }} \) expanded by an automorphism,” Vestnik NGU, Mat., Mekh., Inf., 10, No. 3, 110-118 (2010).zbMATHGoogle Scholar
  15. 15.
    N. A. Bazhenov and R. R. Tukhbatullina, “Constructivizability of the Boolean algebra \( \mathfrak{B}\left( \omega \right) \) with a distinguished automorphism,” Algebra Logika, 51, No. 5, 579-607 (2012).CrossRefGoogle Scholar
  16. 16.
    S. S. Goncharov, Countable Boolean Algebras and Decidability, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1996).Google Scholar
  17. 17.
    S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar
  18. 18.
    C. J. Ash and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Stud. Logic Found. Math., 144, Elsevier, Amsterdam (2000).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia
  2. 2.CERGE–EI, a joint workplace of Charles Univ. and Economics Inst. Acad. Sci. Czech Repub.PragueCzech Republic

Personalised recommendations