A uniform refinement property for congruence lattices
Friedrich Wehrung · Proceedings of the American Mathematical Society · 1999
The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt’s construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tůma. In a previous paper, we constructed a distributive algebraic lattice $A$ with $\aleph _2$ compact elements that cannot be obtained by Schmidt’s construction. In this paper, we show that the same lattice $A$ cannot be obtained using the Pudlák, Tischendorf, Tůma approach. The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.