Journal of the Australian Mathematical Society (Series A)

Research Article

On congruence lattices of m-complete lattices

G. Grätzera1 and H. Laksera1

a1 University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

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.

(Received February 12 1990)

1991 Mathematics subject classification (Amer. Math. Soc.)

  • 08 A 65;
  • 08 A 30

Keywords and phrases

  • Complete lattice;
  • m-complete lattice;
  • complete congruence;
  • m-complete congruence;
  • congruence lattice;
  • complete congruence lattice;
  • automorphism group