Congruence-preserving subdirect products

An algebra A is said to be a congruence-preserving extension of a subalgebra B if the mapping from the congruence lattice of B to that of A, assigning to each congruence relation β on B the minimal congruence relation on A containing β, is an isomorphism. We give a necessary and sufficient condition on the congruence lattice of a subdirect product B of finitely many algebras in a congruence-distributive variety that the full direct product be a congruence-preserving extension of B. We give several applications to congruence lattices of lattices. Received May 25, 2000; accepted in final form January 22, 2001.