Algebra and Logic

, Volume 52, Issue 5, pp 355–366 | Cite as

Computable Numberings of the Class of Boolean Algebras with Distinguished Endomorphisms

  • N. A. BazhenovEmail author

We deal with computable Boolean algebras having a fixed finite number λ of distinguished endomorphisms (briefly, E λ -algebras). It is shown that the index set of E λ -algebras is \( \Pi_2^0 \)-complete. It is proved that the class of all computable E λ -algebras has a \( \Delta_3^0 \)-computable numbering but does not have a \( \Delta_2^0 \)-computable numbering up to computable isomorphism. Also for the class of all computable E λ -algebras, we explore whether there exist hyperarithmetical Friedberg numberings up to \( \Delta_{\alpha}^0 \)-computable isomorphism.


computable Boolean algebra with distinguished endomorphisms computable numbering Friedberg numbering index set isomorphism problem 


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  1. 1.
    S. S. Goncharov and J. F. Knight, “Computable structure and non-structure theorems,” Algebra Logika, 41, No. 6, 639-681 (2002).zbMATHMathSciNetGoogle Scholar
  2. 2.
    V. P. Dobritsa, “The complexity of the index set of a constructive model,” Algebra Logika, 22, No. 4, 269-276 (1983).CrossRefMathSciNetGoogle Scholar
  3. 3.
    W. White, “On the complexity of categoricity in computable structures,” Math. Log. Quart., 49, No. 6, 603-614 (2003).CrossRefzbMATHGoogle Scholar
  4. 4.
    W. Calvert, D. Cummins, J. F. Knight, and S. Miller, “Comparing classes of finite structures,” Algebra Logika, 43, No. 6, 666-701 (2004).zbMATHMathSciNetGoogle Scholar
  5. 5.
    W. Calvert, “The isomorphism problem for classes of computable fields,” Arch. Math. Log., 43, No. 3, 327-336 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    W. Calvert, “The isomorphism problem for computable Abelian p-groups of bounded length,” J. Symb. Log., 70, No. 1, 331-345 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    W. Calvert and J. F. Knight, “Classification from a computable viewpoint,” Bull. Symb. Log., 12, No. 2, 191-218 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    W. Calvert, V. S. Harizanov, J. F. Knight, and S. Miller, “Index sets of computable structures,” Algebra Logika, 43, No. 5, 538-574 (2006).MathSciNetGoogle Scholar
  9. 9.
    W. Calvert, E. Fokina, S. S. Goncharov, J. F. Knight, O. Kudinov, A. S. Morozov, and V. Puzarenko, “Index sets for classes of high rank structures,” J. Symb. Log., 72, No. 4, 1418-1432 (2007).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    E. B. Fokina, “Index sets of decidable models,” Sib. Mat. Zh., 48, No. 5, 1167-1179 (2007).zbMATHMathSciNetGoogle Scholar
  11. 11.
    E. N. Pavlovskii, “Estimation of the algorithmic complexity of classes of computable models,” Sib. Mat. Zh., 49, No. 3, 635-649 (2008).MathSciNetGoogle Scholar
  12. 12.
    E. B. Fokina, “Index sets for some classes of structures,” Ann. Pure Appl. Log., 157, Nos. 2/3, 139-147 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    B. F. Csima, A. Montalbán, and R. A. Shore, “Boolean algebras, Tarski invariants, and index sets,” Notre Dame J. Formal Log., 47, No. 1, 1-23 (2006).CrossRefzbMATHGoogle Scholar
  14. 14.
    N. T. Kogabaev, “Complexity of some natural problems on the class of computable I-algebras,” Sib. Math. Zh., 47, No. 2, 352-360 (2006).zbMATHMathSciNetGoogle Scholar
  15. 15.
    S. S. Goncharov, “Autostability of models and Abelian groups,” Algebra Logika, 19, No. 1, 23-44 (1980).MathSciNetGoogle Scholar
  16. 16.
    S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar
  17. 17.
    C. J. Ash and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Stud. Log. Found. Math., 144, Elsevier, Amsterdam (2000).zbMATHGoogle Scholar
  18. 18.
    S. S. Goncharov, Countable Boolean Algebras and Decidability, Siberian School of Algebra and Logic [in Russian], Nauch. Kniga, Novosibirsk (1996).Google Scholar
  19. 19.
    N. T. Kogabaev, “Universal numbering for constructive I-algebras,” Algebra Logika, 40, No. 5, 561-579 (2001).zbMATHMathSciNetGoogle Scholar
  20. 20.
    N. T. Kogabaev, “The class of projective planes is noncomputable,” Algebra Logika, 47, No. 4, 428-455 (2008).zbMATHMathSciNetGoogle Scholar
  21. 21.
    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
  22. 22.
    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).MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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