Abstract
Let S be a distributive {?, 0}-semilattice. In a previous paper, the second author proved the following result: Suppose that S is a lattice. Let K be a lattice, let $\varphi$: Con K $\to$ S be a {?, 0}-homomorphism. Then $\varphi$ is, up to isomorphism, of the form Conc f, for a lattice L and a lattice homomorphism f : K $\to$ L. In the statement above, Conc K denotes as usual the {?, 0}-semilattice of all ?nitely generated congruences of K. We prove here that this statement characterizes S being a lattice.
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