The Complete Congruence Lattice of a Complete Lattice

  • G. Grätzer


G. Birkhoff [1] raised the following question in 1945: Is every complete lattice isomorphic to the lattice of congruence relations of a suitable (infinitary) algebra? In 1948, Birkhoff restated this question in the Second Edition of his Lattice Theory [2]; however, “(infinitary)” was dropped from the question. This was intentional; G. Birkhoff referred to some continuity conditions that must hold in a congruence lattice of a (finitary) algebra.


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  1. 1.
    G. Birkhoff, Universal Algebra, in “Proc. First Canadian Math. Congress, Montreal, 1945”, University of Toronto Press, Toronto, 1946, pp. 310–326.Google Scholar
  2. 2.
    —, “Lattice Theory”, Amer. Math. Soc. Colloq. Publ. vol. 25, revised edition, Amer. Math. Soc, New York, N.Y., 1948.Google Scholar
  3. 3.
    R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compos. Math. 6 (1938), 239–250.MathSciNetGoogle Scholar
  4. 4.
    —, Lattices with a given group of automorphisms, Canad. J. Math. 2 (1950), 417–419.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    G. Grätzer, “General Lattice Theory”, Academic Press, New York, N.Y.; Birkhäuser Verlag, Basel; Akademie Verlag, Berlin, 1978.CrossRefGoogle Scholar
  6. 6.
    —, “Universal Algebra. Second Edition”, Springer Verlag, New York, Heidelberg, Berlin, 1979.Google Scholar
  7. 7.
    —, On the automorphism group and the complete congruence lattice of a complete lattice, Abstracts of papers presented to the Amer. Math. Soc. 88T-06-215.Google Scholar
  8. 8.
    G. Grätzer, H. Lakser, and B. Wolk, On the lattice of complete congruences of a complete lattice: On a result of K. Reuter and R. Wille, Preprint. University of Manitoba (1988), 1-8.Google Scholar
  9. 9.
    G. Grätzer and W. A. Lampe, Representations of complete lattices as congruence lattices of infinitary algebras. I., II., III. Abstracts, Notices Amer. Math. Soc. 18, 19 (1971-1972), 937, A-683, A-749.Google Scholar
  10. 10.
    G. Grätzer and E. T. Schmidt, On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13 (1962), 179–185.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    —, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59.MathSciNetzbMATHGoogle Scholar
  12. 12.
    W. A. Lampe, On the congruence lattice characterization theorem, Trans. Amer. Math. Soc. 182 (1973), 43–60.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    P. Pudlák, A new proof of the congruence lattice representation theorem, Algebra Universalis 6 (1976), 269–275.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. Pultr and V. Trnková, “Combinatorial algebraic and topological representations of groups, semigroups and categories”, Academia, Prague, 1980.zbMATHGoogle Scholar
  15. 15.
    K. Reuter and R. Wille, Complete congruence relations of complete lattices, Acta Sci. Math. (Szeged) 51 (1987), 319–327.MathSciNetzbMATHGoogle Scholar
  16. 16.
    G. Sabidussi, Graphs with given infinite groups, Monatsch. Math. 68 (1960), 64–67.MathSciNetCrossRefGoogle Scholar
  17. 17.
    E. T. Schmidt, “Kongruenzrelationen algebraischer Strukturen,” Math. Forschungberichte, XXV. VEB Deutcher Verlag der Wissenschaften, Berlin, 1967.Google Scholar
  18. 18.
    S.-K. Teo, Representing finite lattices as complete congruence lattices of complete lattices, Abstracts of papers presented to the Amer. Math. Soc. 88T-06-207.Google Scholar
  19. 19.
    R. Wille, Subdirect decompositions of concept lattices, Algebra Universalis 17 (1983), 275–287.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • G. Grätzer
    • 1
  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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