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Semilattices with the strong endomorphism kernel property. (English) Zbl 1305.06004

By a strong endomorphism on an algebra \(A\) is meant an endomorphism which is compatible with every congruence on \(A\). An algebra \(A\) has the strong endomorphism kernel property if every congruence on \(A\) other than the universal congruence, is the kernel of a strong endomorphism on \(A\). There are characterized strong endomorphisms on meet semilattices and it is shown that a semilattice has the strong endomorphism kernel property if and only if either it is of length 1 or it has length 2 and contains a monolith.

MSC:

06A12 Semilattices
08A35 Automorphisms and endomorphisms of algebraic structures
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