Remarks on homomorphism-homogeneous lattices and semilattices

We consider lattices and semilattices enjoying the homomorphism-homogeneity property introduced recently by P. J. Cameron and J. Nešetřil. First we completely characterize all homomorphism-homogeneous lattices. Also, as a consequence of some general results, we exhibit transparent examples of semilattices both with and without this property. Finally, we show that the endomorphism monoid of Ω, the (unique) countable universal homogeneous semilattice, embeds every finite semigroup.