Algebras with Restricted Cardinalities of Congruence Classes

Let A = (A;F ) be an algebra and ConA its congruence lattice. Recall that A is congruence uniform (see e.g. [1], [5]) if for each Θ ∈ ConA and every a, b ∈ A, card[a]Θ = card[b]Θ. Examples of congruence uniform algebras are e.g. groups, rings or Boolean algebras. A variety V is congruence uniform if each A ∈ V has this property. It was proved by W. Taylor [5] that every congruence uniform variety is congruence regular (see e.g. [1], [3]). The problem if this assertion remains true for a single algebra was solved in [2] and, in a more advanced version, also by M. Goldstern [4]. The following was proved (see [2])