  A variety V has Boolean factor congruences (BFC) if the set of factor
congruences of every algebra in V 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 R. Willard's work ("Varieties Having Boolean
Factor Congruences," J. Algebra, 132 (1990)).
