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On congruence lattices of m-complete lattices

Part of: Algebraic structures

Published online by Cambridge University Press:  09 April 2009

G. Grätzer  and
H. Lakser
G. Grätzer
Affiliation:
University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
H. Lakser
Affiliation:
University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
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Abstract

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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.

Keywords

Complete latticem-complete latticecomplete congruencem-complete congruencecongruence latticecomplete congruence latticeautomorphism group

MSC classification

Secondary: 08A65: Infinitary algebras 08A30: Subalgebras, congruence relations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 1 , February 1992 , pp. 57 - 87
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Frucht, R., ‘Herstellung von Graphen mit vorgegebener abstrakter Gruppe’, Compos. Math. 6 (1938), 239–250.Google Scholar
[2]Frucht, R., ‘Lattices with a given group of automorphisms’, Canad. J. Math. 2 (1950), 417–419.CrossRefGoogle Scholar
[3]Grätzer, G., General Lattice Theory, Academic Press, New York, N. Y., 1978; Birkhäuser Verlag, Basel;Akademie Verlag, Berlin.CrossRefGoogle Scholar
[4]Grätzer, G., Universal Algebra, Second Edition, Springer-Verlag, New York, Heidelberg, Berlin, 1979.CrossRefGoogle Scholar
[5]Grätzer, G., ‘On the 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
[6]Grätzer, G., ‘A “lattice theoretic” proof of the independence of the authomorphism group, the congruence lattice, and subalgebra lattice of an infinitary algebra’, Abstracts of papers presented to the Amer. Math. Soc. 88T-08–254; Algebra Universalis (to appear).Google Scholar
[7]Grätzer, G. and Lakser, H., ‘On the m-complete congruence lattice and the automorphism group of an m-complete lattice’, Abstracts of papers presented to the Amer. Math. Soc. 88T-06–253.Google Scholar
[8]Grätzer, G. and Lakser, H., ‘On complete congruence lattics of complete lattics’, Trans. Amer. Math. Soc. (to appear).Google Scholar
[9]Grätzer, G. and Lakser, H., ‘Congruence lattices of planar lattices’, Abstracts of papers presented to the Amer. Math. Soc. 89T-06–29; Acta Math. Hungar. (to appear).Google Scholar
[10]Grätzer, G. and Lakser, H., ‘Homomorphisms of distributive lattices as restrictions of congruences. II. Restrictions of automorphisms’, Abstracts of papers presented to the Amer. Math. Soc. 89T-06–92; Manuscript. University of Manitoba (1989).Google Scholar
[11]Grätzer, G. and Lakser, H., ‘Congruence lattices, automorphism groups of finite lattices and planarity’, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 137–142.Google Scholar
[12]Pultr, A. and Trnková, V., Combinatorial, algebraic and topological representations of groups, semigroups and categories, Academia, Prague, 1980.Google Scholar
[13]Sabidussi, G., ‘Graphs with given infinite groups’, Monatsh. Math. 64 (1960), 64–67.CrossRefGoogle Scholar
[14]Teo, S.-K., ‘Representing finite lattices as complete congruence lattices of complete lattices’, Abstracts of papers presented to the Amer. Math. Soc. 88T-06–207; Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (to appear).Google Scholar