Congruences of Clone Lattices, II

We continue the study of congruences of clone lattices ℒA, where A is finite, started in an earlier paper by the author and A. P. Semigrodskikh. We prove that each clone that either contains all unary operations or consists of essentially unary operations forms a one-element class of any non-trivial congruence of ℒA. As a consequence, we get that ℒA has the greatest non-trivial congruence provided the lattice is not simple, that ℒA is directly indecomposable, and that it has neither distributive nor dually distributive elements except for the trivial ones.