Abstract
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.
This research was supported by the NSERC of Canada.
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References
G. Birkhoff, Universal Algebra, in “Proc. First Canadian Math. Congress, Montreal, 1945”, University of Toronto Press, Toronto, 1946, pp. 310–326.
—, “Lattice Theory”, Amer. Math. Soc. Colloq. Publ. vol. 25, revised edition, Amer. Math. Soc, New York, N.Y., 1948.
R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compos. Math. 6 (1938), 239–250.
—, Lattices with a given group of automorphisms, Canad. J. Math. 2 (1950), 417–419.
G. Grätzer, “General Lattice Theory”, Academic Press, New York, N.Y.; Birkhäuser Verlag, Basel; Akademie Verlag, Berlin, 1978.
—, “Universal Algebra. Second Edition”, Springer Verlag, New York, Heidelberg, Berlin, 1979.
—, 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.
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.
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.
G. Grätzer and E. T. Schmidt, On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13 (1962), 179–185.
—, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59.
W. A. Lampe, On the congruence lattice characterization theorem, Trans. Amer. Math. Soc. 182 (1973), 43–60.
P. Pudlák, A new proof of the congruence lattice representation theorem, Algebra Universalis 6 (1976), 269–275.
A. Pultr and V. Trnková, “Combinatorial algebraic and topological representations of groups, semigroups and categories”, Academia, Prague, 1980.
K. Reuter and R. Wille, Complete congruence relations of complete lattices, Acta Sci. Math. (Szeged) 51 (1987), 319–327.
G. Sabidussi, Graphs with given infinite groups, Monatsch. Math. 68 (1960), 64–67.
E. T. Schmidt, “Kongruenzrelationen algebraischer Strukturen,” Math. Forschungberichte, XXV. VEB Deutcher Verlag der Wissenschaften, Berlin, 1967.
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.
R. Wille, Subdirect decompositions of concept lattices, Algebra Universalis 17 (1983), 275–287.
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Grätzer, G. (1990). The Complete Congruence Lattice of a Complete Lattice. In: Almeida, J., Bordalo, G., Dwinger, P. (eds) Lattices, Semigroups, and Universal Algebra. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2608-1_9
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