An endomorphism of an algebra A is said to be strong if it is compatible
with every congruence on A; and A is said to have the strong endomorphism kernel
property if every congruence on A, other than the universal congruence, is the kernel
of a strong endomorphism on A. Here we describe, by way of Priestley duality, the
distributive double p-algebras that have the strong endomorphism kernel property.