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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)
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