Journal of Mathematical Sciences

, Volume 215, Issue 4, pp 460–474 | Cite as

Boolean Algebras with Distinguished Endomorphisms and Generating Trees

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We characterize computable Boolean algebras with distinguished endomorphisms in terms of generating trees and mappings of these trees. We show that every degree spectrum of a countable family of subsets of ω is the degree spectrum of some natural enrichment of a Boolean algebra. Bibliography: 20 titles.

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References

  1. 1.
    S. S. Goncharov, “Constructivizability of superatomic Boolean algebras” [in Russian], Algebra Logika 12, No. 1, 31–40 (1973); English transl.: Algebra Logic 12, No. 6, 17–22 (1974).Google Scholar
  2. 2.
    N. T. Kogabaev, “Universal numbering for constructive I-algebras” [in Russian], Algebra Logika 40, No. 5, 561–579 (2001); English transl.: Algebra Logic 40, No. 5, 315–326 (2001).Google Scholar
  3. 3.
    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
  4. 4.
    Yu. L. Ershov, Theory of Numberings [in Russian], Nauka, Moscow (1977).Google Scholar
  5. 5.
    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
  6. 6.
    Yu. L. Ershov and S. S. Goncharov, Constructive Models, [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1999); English transl.: Consultants Bureau, New York (2000).Google Scholar
  7. 7.
    C. J. Ash and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Elsevier, Amsterdam (2000).MATHGoogle Scholar
  8. 8.
    N. A. Bazhenov, “Computable numberings of the class of Boolean algebras with distinguished endomorphisms” [in Russian], Algebra Logika 52, No. 5, 535–552 (2013); English transl.: Algebra Logic 52, No. 5, 355–366 (2013).Google Scholar
  9. 9.
    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).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    S. S. Goncharov, “Computable single-valued numerations” [in Russian], Algebra Logika 19, No. 5, 507–551 (1980); English transl.: Algebra Logic 19, No. 5, 325–356 (1981).Google Scholar
  11. 11.
    J. F. Knight, “Degrees coded in jumps of orderings,” J. Symb. Log. 51, No. 4, 1034–1042 (1986).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    S. S. Goncharov, “Problem of the number of non-self-equivalent constructivizations” [in Russian], Algebra Logika 19, No. 6, 621–639 (1980); English transl.: Algebra Logic 19, No. 6, 401–414 (1980).Google Scholar
  13. 13.
    I. Sh. Kalimullin, “Spectra of degrees of some structures” [in Russian], Algebra Logika 46, No. 6, 729–744 (2007); English transl.: Algebra Logic 46, No. 6, 399–408 (2007).Google Scholar
  14. 14.
    I. Sh. Kalimullin, “Almost computably enumerable families of sets” [in Russian], Mat. Sb. 199, No. 10, 33–40 (2008); English transl.: Sb. Math. 199, No. 10, 1451–1458 (2008).Google Scholar
  15. 15.
    S. Wehner, “Enumerations, countable structures and Turing degrees” Proc. Am. Math. Soc. 126, No. 7, 2131–2139 (1998).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    I. Sh. Kalimullin, Degree Spectra of Algorithmic Systems [in Russian], KFU, Kazan (2013).Google Scholar
  17. 17.
    N. A. Bazhenov and R. R. Tukhbatullina, “Computable categoricity of the Boolean algebra ℬ(ω) with a distinguished automorphism” [in Russian], Algebra Logika 52, No. 2, 131–144 (2013); English transl.: Algebra Logic 52, No. 2, 89–97 (2013).Google Scholar
  18. 18.
    P. M. Semukhin, “The degree spectra of definable relations on Boolean algebras” [in Russian], Sib. Mat. Zh. 46, No. 4, 928–941 (2005); English transl.: Sib. Mat. J. 46, No. 4, 740–750 (2005).Google Scholar
  19. 19.
    T. A. Slaman, “Relative to any nonrecursive set,” Proc. Am. Math. Soc. 126, No. 7, 2117–2122 (1998).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    B. M. Khoussainov and T. M. Kowalski, “Computable isomorphisms of Boolean algebras with operators” Stud. Log. 100, No. 3, 481–496 (2012).MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

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

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