Siberian Mathematical Journal

, Volume 58, Issue 6, pp 959–970 | Cite as

Computability of Distributive Lattices

  • N. A. Bazhenov
  • A. N. Frolov
  • I. Sh. Kalimullin
  • A. G. Melnikov


The class of (not necessarily distributive) countable lattices is HKSS-universal, and it is also known that the class of countable linear orders is not universal with respect to degree spectra neither to computable categoricity. We investigate the intermediate class of distributive lattices and construct a distributive lattice with degree spectrum {d: d ≠ 0}. It is not known whether a linear order with this property exists. We show that there is a computably categorical distributive lattice that is not relatively Δ20-categorical. It is well known that no linear order can have this property. The question of the universality of countable distributive lattices remains open.


distributive lattice computable structure degree spectrum computable categoricity 


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  1. 1.
    Knight J. F. and Stob M., “Computable Boolean algebras,” J. Symb. Log., vol. 65, no. 4, 1605–1623 (2000).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Melnikov A. G., “New degree spectra of abelian groups,” Notre Dame J. Form. Log., vol. 58, no. 4, 507–525 (2017).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hirschfeldt D. R., Khoussainov B., Shore R. A., and Slinko A. M., “Degree spectra and computable dimensions in algebraic structures,” Ann. Pure Appl. Logic, vol. 115, no. 1–3, 71–113 (2002).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fokina E. B. and Friedman S.-D., “Equivalence relations on classes of computable structures,” in: Mathematical Theory and Computational Practice, Springer-Verlag, Berlin, 2009, 198–207 (Lect. Notes Comput. Sci.; vol. 5635).CrossRefGoogle Scholar
  5. 5.
    Harrison-Trainor M., Melnikov A., Miller R., and Montalbán A., “Computable functors and effective interpretability,” J. Symb. Log., vol. 82, no. 1, 77–97 (2017).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Montalbán A., “Computability theoretic classifications for classes of structures,” in: Proc. Intern. Congr. Mathematicians (Seoul, 2014), Kyung Moon Sa Co., Seoul, 2014, vol. II, 79–101.Google Scholar
  7. 7.
    Downey R. G., “Computability theory and linear orderings,” in: Handbook of Recursive Mathematics, Elsevier, Amsterdam, 1998, vol. 2, 823–976.Google Scholar
  8. 8.
    Frolov A., Harizanov V., Kalimullin I., Kudinov O., and Miller R., “Spectra of highn and non-lown degrees,” J. Log. Comput., vol. 22, no. 4, 755–777 (2012).CrossRefMATHGoogle Scholar
  9. 9.
    Richter L. J., “Degrees of structures,” J. Symb. Log., vol. 46, no. 4, 723–731 (1981).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Miller R., “The Δ2 0-spectrum of a linear order,” J. Symb. Log., vol. 66, no. 2, 470–486 (2001).CrossRefGoogle Scholar
  11. 11.
    Goncharov S. S. and Dzgoev V. D., “Autostability of models,” Algebra and Logic, vol. 19, no. 1, 28–37 (1980).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Remmel J. B., “Recursively categorical linear orderings,” Proc. Amer. Math. Soc., vol. 83, no. 2, 387–391 (1981).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ash C. J., “Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees,” Trans. Amer. Math. Soc., vol. 298, no. 2, 497–514 (1986).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    McCoy C. F. D., “Δ2 0-Categoricity in Boolean algebras and linear orderings,” Ann. Pure Appl. Logic, vol. 119, no. 1–3, 85–120 (2003).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    McCoy Ch. F. D., “Partial results in Δ3 0-categoricity in linear orderings and Boolean algebras,” Algebra and Logic, vol. 41, no. 5, 295–305 (2002).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Frolov A. N., “Effective categoricity of computable linear orderings,” Algebra and Logic, vol. 54, no. 5, 415–417 (2015).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Selivanov V. L., “Algorithmic complexity of algebraic systems,” Math. Notes, vol. 44, no. 6, 944–950 (1988).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Turlington A., Computability of Heyting Algebras and Distributive Lattices, Ph. D. Thesis, Univ. of Connecticut (2010).Google Scholar
  19. 19.
    Bazhenov N. A., “Effective categoricity for distributive lattices and Heyting algebras,” Lobachevskii J. Math., vol. 38, no. 4, 600–614 (2017).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wehner S., “Enumerations, countable structures and Turing degrees,” Proc. Amer. Math. Soc., vol. 126, no. 7, 2131–2139 (1998).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Knight J. F., “Degrees coded in jumps of orderings,” J. Symb. Log., vol. 51, no. 4, 1034–1042 (1986).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Slaman T. A., “Relative to any nonrecursive set,” Proc. Amer. Math. Soc., vol. 126, no. 7, 2117–2122 (1998).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • N. A. Bazhenov
    • 1
  • A. N. Frolov
    • 2
  • I. Sh. Kalimullin
    • 2
  • A. G. Melnikov
    • 3
  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  2. 2.Kazan (Volga Region) Federal UniversityKazanRussia
  3. 3.Massey UniversityMasseyNew Zealand

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