Algebra and Logic

, Volume 57, Issue 6, pp 445–462 | Cite as

Structure of Quasivariety Lattices. I. Independent Axiomatizability

  • A. V. KravchenkoEmail author
  • A. M. Nurakunov
  • M. V. Schwidefsky

We find a sufficient condition for a quasivariety K to have continuum many subquasivarieties that have no independent quasi-equational bases relative to K but have ω-independent quasi-equational bases relative to K. This condition also implies that K is Q-universal.


independent basis quasi-identity quasivariety quasivariety lattice Q-universality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Birkhoff, “Universal algebra,” Proc. of the First Canadian Mathematical Congress (Montreal, 1945), Univ. of Toronto, Toronto (1946), pp. 310-326.Google Scholar
  2. 2.
    A. I. Mal’tsev, “Borderline problems of algebra and logic,” Proc. of the Int. Math. Congress (Moscow, 1966), Mir, Moscow (1968), pp. 217-231.Google Scholar
  3. 3.
    V. P. Belkin and V. A. Gorbunov, “Filters in lattices of quasivarieties of algebraic systems,” Algebra and Logic, 14, No. 4, 229–239 (1975).CrossRefzbMATHGoogle Scholar
  4. 4.
    A. I. Budkin, “Filters in lattices of quasivarieties of groups,” Math. USSR–Izv., 33, No. 1, 201-207 (1989).MathSciNetzbMATHGoogle Scholar
  5. 5.
    V. A. Gorbunov, “Covers in lattices of quasivarieties and independent axiomatizability,” Algebra and Logic, 16, No. 5, 340-369 (1977).CrossRefzbMATHGoogle Scholar
  6. 6.
    A. I. Budkin, “Quasivarieties of groups without covers,” Math. Notes, 37, No. 5, 333-337 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. K. Kartashov, “Lattices of quasivarieties of unars,” Sib. Math. J., 26, No. 3, 346-357 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S. V. Sizyĭ, “Quasivarieties of graphs,” Sib. Math. J., 35, No. 4, 783-794 (1994).Google Scholar
  9. 9.
    M. V. Sapir, “The lattice of quasivarieties of semigroups,” Alg. Univ., 21, Nos. 2/3, 172-180 (1985).Google Scholar
  10. 10.
    M. E. Adams and W. Dziobiak, “Q-universal quasivarieties of algebras,” Proc. Am. Math. Soc., 120, No. 4, 1053-1059 (1994).Google Scholar
  11. 11.
    W. Dziobiak, “On lattice identities satisfied in subquasivariety lattices of varieties of modular lattices,” Alg. Univ., 22, Nos. 2/3, 205-214 (1986).Google Scholar
  12. 12.
    V. A. Gorbunov, “Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II,” Algebra and Logic, 34, No. 4, 203-218 (1995).Google Scholar
  13. 13.
    V. A. Gorbunov, Algebraic Theory of Quasivarieties, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar
  14. 14.
    M. E. Adams and W. Dziobiak, “Finite-to-finite universal quasivarieties are Q-universal,” Alg. Univ., 46, Nos. 1/2, 253-283 (2001).Google Scholar
  15. 15.
    A. M. Nurakunov, “Unreasonable lattices of quasivarieties,” Int. J. Alg. Comput., 22, No. 3 (2012), p. No. 1250006.Google Scholar
  16. 16.
    A. M. Nurakunov, “Quasivariety lattices of pointed Abelian groups,” Algebra and Logic, 53, No. 3, 238-257 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    M. V. Schwidefsky, “On complexity of quasivariety lattices,” Algebra and Logic 54, No. 3, 245-257 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    S. M. Lutsak, “On complexity of quasivariety lattices,” Sib. El. Mat. Izv., 14, 92-74 (2017);
  19. 19.
    A. I. Mal’tsev, “Universally axiomatizable subclasses of locally finite classes of models,” Sib. Math. J., 8, No. 5, 764-770 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Tarski, “Equational logic and equational theories of algebras,” in Contributions to Mathematical Logic (Proc. Logic Colloq., Hannover, 1966), North Holland, Amsterdam (1968), pp. 275-288.Google Scholar
  21. 21.
    A. V. Kravchenko, A. M. Nurakunov, and M. V. Schwidefsky, “Structure of quasivariety lattices. II. Undecidable problems,” to appear in Algebra and Logic.Google Scholar
  22. 22.
    A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).Google Scholar
  23. 23.
    G. A. Fraser and A. Horn, “Congruence relations in direct products,” Proc. Am. Math. Soc., 26, 390-394 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    W. Dziobiak, “Selected topics in quasivarieties of algebraic systems,” manuscript of the lecture notes delivered at the Workshop on Comput. Algebra and Appl. to Semigroup Theory (the Center of Algebra of Lisbon Univ., Portugal, November 17-21, 1997).Google Scholar
  25. 25.
    W. Dziobiak, “Quasivarieties of Sugihara semilattices with involution,” Algebra and Logic, 39, No. 1, 26-36 (2000).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • A. V. Kravchenko
    • 1
    • 2
    • 3
    • 4
    Email author
  • A. M. Nurakunov
    • 5
  • M. V. Schwidefsky
    • 1
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Siberian Institute of ManagementNovosibirskRussia
  4. 4.Novosibirsk State Technical UniversityNovosibirskRussia
  5. 5.Institute of Mathematics, National Academy of Science of the Kyrgyz RepublicBishkekKyrgyzstan

Personalised recommendations