Complete congruence lattices of complete distributive lattices (Q1345914)

The authors deal with the question of whether every complete lattice \(L\) is isomorphic to the lattice of complete congruence relations of a suitable complete lattice \(K\). They prove that \(K\) can always be chosen as a complete distributive lattice. In fact, they prove a more general result: Let \(m\) be a regular cardinal \(>\aleph_ 0\). Every \(m\)-algebraic lattice \(L\) can be represented as the lattice of \(m\)-complete congruence relations of an \(m\)-complete distributive lattice \(K\).