algebra universalis

, 15:195 | Cite as

On the structure of varieties with equationally definable principal congruences I

  • W. J. Blok
  • D. Pigozzi


Distributive Lattice Congruence Lattice Subdirect Product Heyting Algebra Discriminator Variety 


  1. [1]
    R. Balbes and Ph.Dwinger,Distributive Lattices. University of Missouri Press, 1974.Google Scholar
  2. [2]
    K. A. Baker,Equational axioms for classes of Heyting algebras. Algebra Universalis6 (1976), p. 105–120.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    —,Finite equational bases for finite algebras in a congruence-distributive equational class. Advances in Math.24 (1977), p. 207–243.MathSciNetMATHGoogle Scholar
  4. [4]
    —,Primitive satisfaction and equational problems for lattices and other algebras. T.A.M.S.190 (1974), p. 125–150.CrossRefMATHGoogle Scholar
  5. [5]
    J. Berman,Distributive lattices with an additional operation. Aequationes Math.16 (1977), p. 165–171.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    C. Bernardi,On the equational class of diagnoziable algebras. Studia Logica34 (1975), p. 321–331.MathSciNetCrossRefGoogle Scholar
  7. [7]
    W. J. Blok,Varieties of interior algebras. Dissertation, University of Amsterdam, 1976.Google Scholar
  8. [8]
    —,The lattice of modal logics, an algebraic investigation, J.S.L.45 (1980), p. 221–236.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    W. J. Blok and Ph.Dwinger,Equational classes of closure algebras. I. Ind. Math.37 (1975), p. 189–198.MathSciNetGoogle Scholar
  10. [10]
    W. J. Blok andD. Pigozzi,The deduction theorem in algebraic logic. Manuscript.Google Scholar
  11. [11]
    S. Bulman-Fleming andH. Werner,Equational compactness in quasi-primal varieties, Algebra Universalis7 (1977), p. 33–46.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    S. Burris andH. Werner,Sheaf constructions and their elementary properties. T.A.M.S.248 (1979), p. 269–309.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    A. Diego,Sur les algèbres de Hilbert. Collection de Logique Mathematique, Series A, No. 21, Paris 1966.Google Scholar
  14. [14]
    R. Franci,Filtra and ideal classes of universal algebras. Quaderni dell'Istituto di Matematica dell'Universita di Siena (1976).Google Scholar
  15. [15]
    E. Fried, G. Grätzer, andR. Qackenbush,Uniform congruence schemes. Algebra Universalis10 (1980), p. 176–189.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    E. Fried andE. W. Kiss,Connection between the congruence lattices and polynomial properties. Preprint.Google Scholar
  17. [17]
    E. Fried andA. F. Pixley,The dual discriminator function in universal algebra. Acta Univ. Szeged41 (1979), p. 83–100.MathSciNetMATHGoogle Scholar
  18. [18]
    G. Grätzer,Universal Algebra. Springer-Verlag, 1979.Google Scholar
  19. [19]
    T. Hecht andT. Katrinák,Principal congruences of p-algebras and double p-algebras, Proc. Amer. Math. Soc.58 (1976), p. 25–31.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    L. Henkin, J. D. Monk andA. Tarski,Cylindric algebras, Part I. North-Holland Publishing Company, Amsterdam 1971.MATHGoogle Scholar
  21. [21]
    V. A. Jankov,The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures. Sov. Math. Dokl.4 (1963), p.-1203–1204.Google Scholar
  22. [22]
    B. Jónsson,Algebras whose congruence lattices are distributive. Math. Scand.21 (1967), p. 110–121.MathSciNetMATHGoogle Scholar
  23. [23]
    B. Jónsson andA. Tarski,Boolean algebras with operators. Part I. Amer. J. Math.73 (1951) p. 891–939.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    —,Boolean algebras with operators. Part II. Amer. J. Math.74 (1952), p. 127–162.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    T. Katrinák,The structure of distributive double p-algebras. Regularity and congruences. Algebra Universalis3 (1973), p. 238–246.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    P. Köhler,Brouwerian semilattices. Math. Institut, Justus Liebig Universität, Giessen.Google Scholar
  27. [27]
    P. Köhler andD. Pigozzi,Varieties with equationally definable principal congruences. Algebra Universalis,11 (1980), p. 213–219.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    H. Lakser,Principal congruences of pseudo-complemented distributive lattices. Proc. Amer. Math. Soc.37 (1973), p. 32–36.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    —,The structure of pseudo-complemented distributive lattices, I. T.A.M.S.,156 (1971), p. 335–342.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    K. B. Lee,Equational classes of distributive pseudo-complemented lattices Canad. J. Math.22 (1970), p. 881–891.MathSciNetMATHGoogle Scholar
  31. [31]
    J. Łos,Quelques remarques, theorèmes et problèmes sur les classes définissables d'algèbres. Mathematical interpretation of formal systems. North-Holland Publ. Co. Amsterdam, 1955, p. 98–113.Google Scholar
  32. [32]
    S. MacLane,Categories for the Working Mathematician. Springer-Verlag, 1971.Google Scholar
  33. [33]
    R. Magari,Varietà a quozienti filtrali. Ann. Univ. Ferrara, Sez. VII14 (1969), 5–20.MathSciNetMATHGoogle Scholar
  34. [34]
    —,Representation and duality theory for diagonizable algebras. Studia Logica34 (1975), p. 305–313.MathSciNetCrossRefGoogle Scholar
  35. [35]
    R. McKenzie,Equational bases and non-modular lattice varieties. T.A.M.S.174 (1972), p. 1–43.MathSciNetCrossRefGoogle Scholar
  36. [36]
    W. C. Nemitz,Implicative semilattices, T.A.M.S.,117 (1965), p. 128–142.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    Semi-Boolean lattices, Notre Dame J. of Formal Logic10 (1969), p. 128–142.MathSciNetCrossRefGoogle Scholar
  38. [38]
    H. Rasiowa,An algebraic approach to non-classical logics. North-Holland Publ. Co., Amsterdam, 1974.MATHGoogle Scholar
  39. [39]
    W. Rautenberg,Der Verband der normalen und verzweigten Modallogiken. Math. Z.156 (1977), p. 123–140.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    W. Rautenberg,Klassische und Nichtklassische Aussagenlogik. Vieweg, 1979.Google Scholar
  41. [41]
    H. P. Sankappanavar,A characterization of principal congruences of De Morgan algebras and its applications. Math Logic in Latin America, Proc. IV Latin Amer. Symp. Math. Logic, Santiago 1978, p. 341–349. North-Holland Pub. Co., Amsterdam, 1980.Google Scholar
  42. [42]
    A. Tarski,Contributions to the theory of model, III. Ind. Math.17 (1955), p. 56–64.MathSciNetGoogle Scholar
  43. [43]
    H. Ursini,Intuitionistic diagonizable algebras. Algebra Universalis,9 (1979), p. 229–237.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    H. Werner,Discriminator algebras. Studien zur Algebra und ihre Anwendungen 6, Akademie Verlag, Berlin (1978).MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag 1982

Authors and Affiliations

  • W. J. Blok
    • 1
    • 2
  • D. Pigozzi
    • 1
    • 2
  1. 1.Simon Fraser UniversityBurnabyCanada
  2. 2.Iowa State UniversityAmesU.S.A.

Personalised recommendations