Studia Logica

, Volume 92, Issue 1, pp 109–120 | Cite as

Quasivarieties with Definable Relative Principal Subcongruences

  • A. M. Nurakunov
  • M. M. StronkowskiEmail author


For quasivarieties of algebras, we consider the property of having definable relative principal subcongruences, a generalization of the concepts of definable relative principal congruences and definable principal subcongruences. We prove that a quasivariety of algebras with definable relative principal subcongruences has a finite quasiequational basis if and only if the class of its relative (finitely) subdirectly irreducible algebras is strictly elementary. Since a finitely generated relatively congruence-distributive quasivariety has definable relative principal subcongruences, we get a new proof of the result due to D. Pigozzi: a finitely generated relatively congruence-distributive quasivariety has a finite quasi-equational basis.


Quasivariety relatively congruence-distributive quasivariety definable relative principal subcongruences finite quasi-equational basis 

2000 Mathematics Subject Classification

08C15 08A30 08B10 


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  1. 1.
    Baker K.A.: ‘Finite equational bases for finite algebras in congruence-distributive equational classes’. Advances in Math. 24, 207–243 (1977)Google Scholar
  2. 2.
    Baker K.A., Wang J.: ‘Definable principal subcongruences’. Algebra Universalis 47, 145–151 (2002)CrossRefGoogle Scholar
  3. 3.
    Belkin V.P.: ‘Quasi-identities of finite rings and lattices’. Algebra Logic 17, 171–179 (1979)CrossRefGoogle Scholar
  4. 4.
    Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Springer- Verlag, 1981. The Millenium Edition available at
  5. 5.
    Czelakowski J., Dziobiak W.: ‘The parameterized local deduction theorem for quasivarieties of algebras and its application’. Algebra Universalis 35, 373–419 (1996)CrossRefGoogle Scholar
  6. 6.
    Dziobiak W.: ‘Finite bases for finitely generated, relatively congruence distributive quasivarieties’. Algebra Universalis 28, 303–323 (1991)CrossRefGoogle Scholar
  7. 7.
    Dziobiak W., Maróti M., McKenzie R., Nurakunov A.M.: ‘The weak extension property and finite axiomatizability for quasivarieties’. Fund. Math. 202, 199–223 (2009)CrossRefGoogle Scholar
  8. 8.
    Gorbunov, V. A., Algebraicheskaya Teoriya Kvazimnogoobrazij, Nauchnaya Kniga, Novosibirsk, 1999. English transl. Algebraic Theory of Quasivarieties, Consultants Bureau, New York 1998.Google Scholar
  9. 9.
    Jónsson B.: ‘Algebras whose congruence lattices are distributive’. Math. Scand. 21, 110–121 (1967)Google Scholar
  10. 10.
    Jónsson B.: ‘On finitely based varieties of algebras’. Colloq. Math. 42, 255–261 (1979)Google Scholar
  11. 11.
    Mal’cev A.I.: Algebraicheskie Sistemy, Nauka, Moscow, 1970. English transl. Algebraic Systems, Springer-Verlag, New York (1973)Google Scholar
  12. 12.
    Maróti M., McKenzie R.: ‘Finite basis problems and results for quasivarieties’. Studia Logica 78, 293–320 (2004)CrossRefGoogle Scholar
  13. 13.
    McKenzie R.: ‘Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties’. Algebra Universalis 8, 336–348 (1978)CrossRefGoogle Scholar
  14. 14.
    McKenzie, R. N., G. F. McNulty, and W. F. Taylor, Algebras, Lattices, Varieties. Vol. I, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, 1987.Google Scholar
  15. 15.
    Nurakunov A.M.: ‘Characterization of relatively distributive quasivarieties of algebras’. Algebra Logic 29, 451–458 (1990)CrossRefGoogle Scholar
  16. 16.
    Nurakunov A.M.: ‘Quasivarieties of algebras with definable principal congruences’. Algebra Logic 29, 26–34 (1990)CrossRefGoogle Scholar
  17. 17.
    Nurakunov A.M.: ‘Quasi-identities of congruence-distributive quasivarieties of algebras’. Siberian Math. J. 42, 108–118 (2001)CrossRefGoogle Scholar
  18. 18.
    Pałasińska K.: ‘Finite basis theorem for filter-distributive protoalgebraic deductive systems and strict universal horn classes’. Studia Logica 74, 233–273 (2003)CrossRefGoogle Scholar
  19. 19.
    Pigozzi D.: ‘Finite basis theorem for relatively congruence-distributive quasivarieties’. Trans. Amer. Math. Soc. 331, 499–533 (1988)CrossRefGoogle Scholar
  20. 20.
    Willard R.: ‘A finite basis theorem for residually finite, congruence meetsemidistributive varieties’. J. Symbolic Logic 65, 187–200 (2000)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsNational Academy of ScienceBishkekKyrghyz Republic
  2. 2.Faculty of Mathematics and Information SciencesWarsaw University of TechnologyWarsawPoland
  3. 3.Eduard Čech CenterCharles UniversityPragueCzech Republic

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