The strong endomorphism kernel property in Ockham algebras

An endomorphism on an algebra A is said to be "strong" if it is compatible with every congruence on A; and si is said to have the "strong endomorphism kernel property" if every congruence on si, different from the universal congruence, is the kernel of a strong endomorphism on A. Here we consider this property in the context of Ockham algebras. In particular, for those MS-algebras that have this property we describe the structure of their dual space in terms of 1-point compactifications of discrete spaces.