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On congruence lattices of m-complete lattices

  • G. Grätzer (a1) and H. Lakser (a1)
Abstract
Abstract

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruence relations of K. In addition, if L has more than one element, then we show how to construct K so that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.

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[2] R. Frucht , ‘Lattices with a given group of automorphisms’, Canad. J. Math. 2 (1950), 417419.

[3] G. Grätzer , General Lattice Theory, Academic Press, New York, N. Y., 1978; Birkhäuser Verlag, Basel;Akademie Verlag, Berlin.

[4] G. Grätzer , Universal Algebra, Second Edition, Springer-Verlag, New York, Heidelberg, Berlin, 1979.

[12] A. Pultr and V. Trnková , Combinatorial, algebraic and topological representations of groups, semigroups and categories, Academia, Prague, 1980.

[13] G. Sabidussi , ‘Graphs with given infinite groups’, Monatsh. Math. 64 (1960), 6467.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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