The endomorphism kernel property in finite distributive lattices and de Morgan algebras

An algebra A has the endomorphism kernel property if every congruence on A different from the universal congruence is the kernel of an endomorphism on A. We first consider this property when A is a finite distributive lattice, and show that it holds if and only if A is a cartesian product of chains. We then consider the case where A is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property.