Congruence-preserving subdirect products. (Q1771890)

An algebra \(\mathcal A\) is said to be a congruence-preserving extension of its subalgebra \(\mathcal B\) if \(\text{Con}\, \mathcal A\) is isomorphic to \(\text{Con}\, \mathcal B\). The author gives a necessary and sufficient condition on the congruence lattice of a subdirect product \(\mathcal B\) of finitely many algebras in a congruence-distributive variety that the direct product is a congruence-preserving extension of \(\mathcal B\). This result yields several theorems on congruence lattices of lattices including the Duffus-Jónsson-Rival theorem and the Grätzer-Lakser-Schmidt theorem.