Unsolvable one-dimensional lifting problems for congruence lattices of   lattices
  Let S be a distributive {&#8744;, 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 $\phi$: Con K $\to$ S be a {&#8744;, 0}-homomorphism. Then $\phi$ 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 {&#8744;, 0}-semilattice of all &#64257;nitely generated congruences of K. We prove here that this statement characterizes S being a lattice. 
Jiri Tuma (MFF-UK), Friedrich Wehrung (LMNO)