Do we need complete-simple distributive lattices?
A complete lattice is said to be complete-simple if it has no nontrivial complete congruence relations. Let \(L\) be a complete lattice; if \(\theta\) is a complete congruence of \(L\), then \(L/\theta\) is called a complete quotient of \(L\). The authors prove the following result: Let \(C(D)\) be the lattice of all complete congruences of a complete distributive lattice \(D\). Suppose that either (i) \(C(D)\) is a three-element chain, or (ii) \(C(D)\) is the five-element nonmodular lattice. Then \(D\) has an infinite distributive complete-simple lattice as a complete quotient.
scientific article; zbMATH DE number 724108
scientific article