On complete congruence lattices of complete lattices
George Grätzer, H. Lakser · Transactions of the American Mathematical Society · 1991
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 L , we construct a complete lattice K K such that L L is isomorphic to the lattice of complete congruence relations of K K . Regarding K 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 K so that it will also have a prescribed automorphism group.