2005/01/25 by Friedrich Wehrung
Mathematics · #math.GM #msc:06A12 #msc:06B10 #msc:16E50
published as Proceedings of the American Mathematical Society 127, no. 2 (1999) 363--370
arxiv created 2005/01/25 · arxiv updated 2009/12/01
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. Pudlak, M. Tischendorf, and J. Tuma. In a previous paper, we constructed a distributive algebraic lattice A with ℵ_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 Pudlak, Tischendorf, Tuma approach. The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, which 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.