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Journal of Mathematical Sciences

, Volume 203, Issue 4, pp 444–454 | Cite as

Δ 2 0 -Categoricity of Boolean Algebras

  • N. A. Bazhenov
Article

We show that the notions of Δ 2 0 -categoricity and relative Δ 2 0 -categoricity in Boolean algebras coincide. We prove that for every Turing degree d <0′ a computable Boolean algebra is d-computably categorical if and only if it is computably categorical. Bibliography: 21 titles.

Keywords

Boolean Algebra Turing Degree Computable Presentation Computably Categorical Enumerable Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Fröhlich and J. Shepherdson, “Effective procedures in field theory,” Philos. Trans. Roy. Soc. London, Ser. A 248, No. 950, 407–432 (1956).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A. I. Mal’tsev, “Constructive algebras. I” [in Russian], Usp. Mat. Nauk 16, No. 3, 3–60 (1961); English transl.: Russ. Math. Surv. 16, No. 3, 77–129 (1961).CrossRefzbMATHGoogle Scholar
  3. 3.
    S. S. Goncharov and Yu. L. Ershov, Constructive Models [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1999); English transl.: Consultants Bureau, New York (2000).Google Scholar
  4. 4.
    S. S. Goncharov, “Computability and computable models,” In: Mathematical Problems from Applied Logic II, pp. 99–216, Springer, New York (2006).Google Scholar
  5. 5.
    S. S. Goncharov and B. Khoussainov, “Open problems in the theory of constructive algebraic systems,” In: Computability Theory and its Applications, pp. 145–170, Am. Math. Soc., Providence, RI (2000).CrossRefGoogle Scholar
  6. 6.
    S. S. Goncharov, “On the number of nonequivalent constructivizations” [in Russian], Algebra Logika 16, No. 3, 257–282 (1977).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    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).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    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. Logic 74, No. 3, 1047–1060 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    S. S. Goncharov, “Autostability and computable families of constructivizations” [in Russian], Algebra Logika 14, No. 6, 647–680 (1975).CrossRefzbMATHGoogle Scholar
  10. 10.
    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, 28–37 (1980).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    J. B. Remmel, “Recursive isomorphism types of recursive Boolean algebras,” J. Symb. Logic 46, No. 3, 572–594 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ch. McCoy, Δ20-Categoricity in Boolean algebras and linear orderings,” Ann. Pure Appl. Logic 119, No. 1–3, 85–120 (2003).Google Scholar
  13. 13.
    Ch. F. 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
  14. 14.
    C. J. Ash, “Categoricity in hyperarithmetical degrees,” Ann. Pure Appl. Logic 34, No. 1, 1–14 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    N. A. Bazhenov, “Hyperarithmetical categoricity of Boolean algebras of type B(ω α × η)” [in Russian], Vest. Novosib. Gos. Univ. Ser. Mat. 12, No. 3, 35–45 (2012); English transl.: J. Math. Sci., New York 202, No. 1, 40–49 (2014).CrossRefMathSciNetGoogle Scholar
  16. 16.
    S. S. Goncharov, Countable Boolean Algebras and Decidability [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1996); English transl.: Plenum, New York (1997).Google Scholar
  17. 17.
    R. I. Soare, Recursively Enumerable Sets and Degrees, Springer, Berlin etc. (1987).CrossRefGoogle Scholar
  18. 18.
    C. J. Ash and J. F. Khight, Computable Structures and the Hyperarithmetical Hierarchy, Elsevier, Amsterdam (2000).zbMATHGoogle Scholar
  19. 19.
    A. Montalbán, “On the triple jump of the set of atoms of a Boolean algebra,” Proc. Am. Math. Soc. 136, No. 7, 2589–2595 (2008).CrossRefzbMATHGoogle Scholar
  20. 20.
    C. E. Sacks, “Recursive enumerability and the jump operator,” Trans. Am. Math. Soc. 108, No. 2, 223–239 (1963).CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    S. S. Goncharov, “Some properties of the constructivizations of Boolean algelbras” [in Russian], Sib. Mat. Zh. 16, No. 2, 264–278 (1975); English transl.: Sib. Math. J. 16, No. 2, 203–214 (1975).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics SB RASNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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