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Δ 02 -Categoricity of Boolean Algebras

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We show that the notions of Δ 02 -categoricity and relative Δ 02 -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.

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References

  1. A. Fröhlich and J. Shepherdson, “Effective procedures in field theory,” Philos. Trans. Roy. Soc. London, Ser. A 248, No. 950, 407–432 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  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. S. S. Goncharov, “Computability and computable models,” In: Mathematical Problems from Applied Logic II, pp. 99–216, Springer, New York (2006).

    Google Scholar 

  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).

    Chapter  Google Scholar 

  6. S. S. Goncharov, “On the number of nonequivalent constructivizations” [in Russian], Algebra Logika 16, No. 3, 257–282 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. S. S. Goncharov, “Autostability and computable families of constructivizations” [in Russian], Algebra Logika 14, No. 6, 647–680 (1975).

    Article  MATH  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  11. J. B. Remmel, “Recursive isomorphism types of recursive Boolean algebras,” J. Symb. Logic 46, No. 3, 572–594 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  12. Ch. McCoy, Δ 02 -Categoricity in Boolean algebras and linear orderings,” Ann. Pure Appl. Logic 119, No. 1–3, 85–120 (2003).

  13. Ch. F. McCoy, “Partial results in Δ 03 -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).

  14. C. J. Ash, “Categoricity in hyperarithmetical degrees,” Ann. Pure Appl. Logic 34, No. 1, 1–14 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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. R. I. Soare, Recursively Enumerable Sets and Degrees, Springer, Berlin etc. (1987).

    Book  Google Scholar 

  18. C. J. Ash and J. F. Khight, Computable Structures and the Hyperarithmetical Hierarchy, Elsevier, Amsterdam (2000).

    MATH  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  20. C. E. Sacks, “Recursive enumerability and the jump operator,” Trans. Am. Math. Soc. 108, No. 2, 223–239 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, 4, pr. Akad. Koptyuga, Novosibirsk, 630090, Russia

    N. A. Bazhenov

  2. Novosibirsk State University, 2, ul. Pirogova, Novosibirsk, 630090, Russia

    N. A. Bazhenov

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  1. N. A. Bazhenov
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Correspondence to N. A. Bazhenov.

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Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 13, No. 2, 2013, pp. 13–23.

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Bazhenov, N.A. Δ 02 -Categoricity of Boolean Algebras. J Math Sci 203, 444–454 (2014). https://doi.org/10.1007/s10958-014-2148-9

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  • DOI: https://doi.org/10.1007/s10958-014-2148-9

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