Varieties Having Boolean Factor Congruences

Every ring R with identity satisfies the following property: the factor ideals of R (i.e., those ideals I such that I+ J= R and In J= (0) for some ideal J) form a Boolean sublattice of the lattice of all ideals of R. The universal algebraic abstraction of this property is known as Boolean factor congruences (BFC) or as the strict refinement property; more examples of algebras having BFC are lattices, semilattices, and centerless groups. We take up the study of varieties all of whose members have BFC, and show that all known examples of such varieties have a first-order definable 4-ary relation witnessing BFC. We also show that if every member of a variety is centerless then the variety has BFC. but not vice versa; and that, for a certain class of varieties, BFC is equivalent to the absence of abelian algebras. 6 1990 Academic Press. Inc.