Algebra and Logic

, Volume 51, Issue 5, pp 384–403 | Cite as

Constructivizability of the boolean algebra \( \mathfrak{B}\left( \omega \right) \) with a distinguished automorphism

Article

A constructivizability criterion for the Boolean algebra \( \mathfrak{B}\left( \omega \right) \) with a distinguished automorphism is given. As a consequence of the criterion, combined with a result due to I. Sh. Kalimullin, B. M. Khoussainov, and A. G. Melnikov, we construct a Boolean algebra with a distinguished automorphism whose degree spectrum contains every nonzero Turing \( \Delta_2^0 \) -degree but does not contain 0.

Keywords

Boolean algebra with distinguished automorphism constructivizability degree spectra of structures 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. S. Goncharov, “Constructivizability of superatomic Boolean algebras,” Algebra Logika, 12, No. 1, 31–40 (1973).MATHCrossRefGoogle Scholar
  2. 2.
    N. T. Kogabaev, “Universal numbering for constructive I-algebras,” Algebra Logika, 40, No. 5, 561–579 (2001).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    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).MATHGoogle Scholar
  4. 4.
    S. S. Goncharov, Countable Boolean Algebras and Decidability, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1996).Google Scholar
  5. 5.
    S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar
  6. 6.
    C. J. Ash and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Stud. Log. Found. Math., 144, Elsevier, Amsterdam (2000).Google Scholar
  7. 7.
    W. Calvert, D. Cenzer, V. Harizanov, and A. Morozov, “Effective categoricity of equivalence structures,” Ann. Pure Appl. Log., 141, Nos. 1/2, 61–78 (2006).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    R. G. Downey, A. M. Kach, and D. Turetsky, “Limitwise monotonic functions and their applications,” Proc. Eleventh Asian Log. Conf. (2011), pp. 59–85.Google Scholar
  9. 9.
    A. M. Kach and D. Turetsky, “Limitwise monotonic functions, sets and degrees on computable domains,” J. Symb. Log., 75, No. 1, 131–154 (2010).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    K. Harris, “η-Representation of sets and degrees,” J. Symb. Log., 73, No. 4, 1097–1121 (2008).MATHCrossRefGoogle Scholar
  11. 11.
    A. M. Kach, “Computable shuffle sums of ordinals,” Arch. Math. Log., 47, No. 3, 211–219 (2008).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    I. Kalimullin, B. Khoussainov, and A. Melnikov, “Limitwise monotonic sequences and degree spectra of structures,” to appear in Proc. Am. Math. Soc. Google Scholar
  13. 13.
    J. F. Knight, “Degrees coded in jumps of orderings,” J. Symb. Log., 51, No. 4, 1034-1042 (1986).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

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

Personalised recommendations