Non-representable distributive semilattices

Abstract

We present two examples of distributive algebraic lattices which are not isomorphic to the congruence lattice of any lattice. The first such example was discovered by F. Wehrung in 2005. One of our examples is defined topologically, the other one involves majority algebras. In particular, we prove that the congruence lattice of the free majority algebra on (at least) ℵ 2 generators is not isomorphic to the congruence lattice of any lattice. Our method is a generalization of Wehrung’s approach, so that we are able to apply it to a larger class of distributive semilattices.