## The strong endomorphism kernel property in distributive double $$p$$-algebras.(English)Zbl 1320.06009

Summary: An endomorphism of 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 $$\mathcal A$$, other than the universal congruence, is the kernel of a strong endomorphism on $$\mathcal A$$. Here we describe, by way of Priestley duality, the distributive double $$p$$-algebras that have the strong endomorphism kernel property.

### MSC:

 06D15 Pseudocomplemented lattices 08A30 Subalgebras, congruence relations 08A35 Automorphisms and endomorphisms of algebraic structures 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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