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On complete congruence lattices of complete lattices

Authors: G. Grätzer and H. Lakser
Journal: Trans. Amer. Math. Soc. 327 (1991), 385-405
MSC: Primary 06B10; Secondary 06A23, 08A30
MathSciNet review: 1036003
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Abstract: The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In this paper, we characterize this lattice as a complete lattice. In other words, for a complete lattice $ L$, we construct a complete lattice $ K$ such that $ L$ is isomorphic to the lattice of complete congruence relations of $ K$. Regarding $ K$ as an infinitary algebra, this result strengthens the characterization theorem of congruence lattices of infinitary algebras of G. Grätzer and W. A. Lampe. In addition, we show how to construct $ K$ so that it will also have a prescribed automorphism group.

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Keywords: Complete lattice, congruence lattice, complete congruence, automorphism group
Article copyright: © Copyright 1991 American Mathematical Society

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