On algebras whose congruences have a class that is a subuniverse

We study algebras and varieties where every non-trivial congruence has some class being a non-trivial subuniverse of the algebra in question. Then we focus on algebras where this non-trivial class is a unique non-singleton class of a congruence. In particular, we investigate Rees algebras, pseudo-Rees algebras and algebras having the one-block property. Many examples are included. At the end, we will also deal with quotient algebras.