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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

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