Evaporation schemes in congruence lattices of majority algebras
We investigate the problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a majority algebra. We try the method used in Wehrung’s solution of Dilworth’s Congruence Lattice Problem. For this purpose we introduce the concept of a strong evaporation scheme in distributive semilattices. The main result says that the semilattices of compact congruences of majority algebras can have strong evaporation schemes of any cardinality. The situation for majority algebras strongly contrasts with the one for lattices, as the latter do not have strong evaporation schemes of cardinality greater than $\aleph _1$. This leaves the original problem open, but we believe that our results and methods are a significant step towards its solution. We also establish a few general results, in particular we prove that distributive semilattices that are lattices can contain strong evaporation schemes of cardinality at most $\aleph _1$.