Complete Congruence Lattices of Complete Distributive Lattices

AbstractIn 1983, Wille raised the question: Is every complete lattice L isomorphic to the lattice of complete congruence relations of a suitable complete lattice K? In 1988, this was answered in the affirmative by the first author. A number of papers have been published on this problem by Freese, Johnson, Lakser, Teo, and the authors. In the present paper we prove that K can always be chosen as a complete distributive lattice. In fact, we prove the following more general result: THEOREM. Let m be a regular cardinal > ℵ0. Every m-algebraic lattice L can be represented as the lattice of m-complete congruence relations of an m-complete distributive lattice K