## The strong endomorphism kernel property in distributive $$p$$-algebras.(English)Zbl 1299.06017

Summary: An endomorphism on an algebra $${\mathcal A}$$ is said to be strong if it is compatible with every congruence on $${\mathcal A}$$; and $${\mathcal A}$$ is said to have the strong endomorphism kernel property if every congruence on $${\mathcal A}$$, except the universal congruence, is a kernel of a strong endomorphism on $${\mathcal A}$$. In this note, we characterize those distributive pseudocomplemented algebras that have the strong endomorphism kernel property. We show that a distributive pseudocomplemented algebra has this property if and only if it is either a 2-element chain, or is of the form $$\{0\}\oplus B$$ where $$B$$ is Boolean.

### MSC:

 06D15 Pseudocomplemented lattices 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)

### Keywords:

strong endomorphism; kernel property; $$p$$-algebra