  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 the
congruence. In particular, we investigate Rees algebras, quasi-Rees algebras
and algebras having the one-block-property. We also present results concerning
these properties on quotient algebras. Several examples of such algebras and
varieties are included.
