Unsolvable one-dimensional lifting problems for congruence lattices of lattices

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 $ϕ$: Con K $\to$ S be a {∨, 0}-homomorphism. Then $ϕ$ 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 finitely generated congruences of K. We prove here that this statement characterizes S being a lattice.