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Siberian Mathematical Journal

, Volume 56, Issue 3, pp 490–498 | Cite as

Autostability relative to strong constructivizations of Boolean algebras with distinguished ideals

  • D. E. Pal’chunovEmail author
  • A. V. Trofimov
  • A. I. Turko
Article

Abstract

We study Boolean algebras with distinguished ideals (I-algebras). We proved that a local I-algebra is autostable relative to strong constructivizations if and only if it is a direct product of finitely many prime models. We describe complete formulas of elementary theories of local Boolean algebras with distinguished ideals and a finite tuple of distinguished constants. We show that countably categorical I-algebras, finitely axiomatizable I-algebras, superatomic Boolean algebras with one distinguished ideal, and Boolean algebras are autostable relative to strong constructivizations if and only if they are products of finitely many prime models.

Keywords

Boolean algebra Boolean algebra with distinguished ideals I-algebra autostability strong constructivizability autostability relative to strong constructivizations prime model 

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References

  1. 1.
    Mal’tsev A. I., “Constructive algebras. I,” Russian Math. Surveys, 16, No. 3, 77–129 (1961).CrossRefzbMATHGoogle Scholar
  2. 2.
    Mal’tsev A. I., “On recursive abelian groups,” Soviet Math., Dokl., 32, 1431–1434 (1962).zbMATHGoogle Scholar
  3. 3.
    Ershov Yu. L., “Constructive models,” in: Selected Problems of Algebra and Logic [in Russian], Nauka, Novosibirsk, 1973, pp. 111–130.Google Scholar
  4. 4.
    Morley M. D., “Decidable models,” Israel J. Math., 25, No. 3–4, 233–240 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ershov Yu. L. and Goncharov S. S., Constructive Models, Ser. Siberian School of Algebra and Logic, Kluwer Academic/Plenum Publishers, New York, etc. (2000).Google Scholar
  6. 6.
    Nurtazin A. T., “Strong and weak constructivization and computable families,” Algebra and Logic, 13, No. 3, 177–184 (1974).CrossRefGoogle Scholar
  7. 7.
    Ershov Yu. L., “Decidability of the elementary theory of distributive structures with relative complements and the theory of filters,” Algebra i Logika, 3, No. 3, 17–38 (1964).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Pal’chunov D. E., “Finitely axiomatizable Boolean algebras with distinguished ideals,” Algebra and Logic, 26, No. 4, 252–266 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pal’chunov D. E., “Countably-categorical Boolean algebras with distinguished ideals,” Studia Logica, 46, No. 2, 121–135 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pal’chunov D. E., “Prime and countably saturated Boolean algebras with distinguished ideals,” Tr. Inst. Mat. (Novosibirsk), 25, 82–103 (1993).MathSciNetGoogle Scholar
  11. 11.
    Pal’chunov D. E., “The theory of Boolean algebras with distinguished ideals without a prime model,” Tr. Inst. Mat. (Novosibirsk), 25, 104–132 (1993).MathSciNetGoogle Scholar
  12. 12.
    Goncharov S. S. and Dzgoev V. D., “Autostability of models,” Algebra and Logic, 19, No. 1, 28–37 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    LaRoche P., “Recursively represented Boolean algebras,” Notices Amer. Math. Soc., 24, A-552 (1977).Google Scholar
  14. 14.
    Kogabaev N. T., “Autostability of Boolean algebras with distinguished ideal,” Siberian Math. J., 39, No. 5, 927–935 (1998).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Alaev P. E., “Autostable I-algebras,” Algebra and Logic, 43, No. 5, 285–306 (2004).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goncharov S. and Khoussainov B., “Open problems in the theory of constructive algebraic systems,” Contemp. Math., 257, 145–170 (2000).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Goncharov S. S., Countable Boolean Algebras and Decidability [in Russian], Nauchnaya Kniga, Novosibirsk (1996).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • D. E. Pal’chunov
    • 1
    Email author
  • A. V. Trofimov
    • 2
  • A. I. Turko
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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