BOOLEAN FACTOR CONGRUENCES AND PROPERTY (*)

A variety [Formula: see text] has Boolean factor congruences (BFC) if the set of factor congruences of any algebra in [Formula: see text] is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal'cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in Willard's work[9].