A uniform refinement property for congruence lattices

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 $\aleph_2$ 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 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.

Mots clés: Semilattices; weakly distributive homomorphisms; congruence splitting lattices; uniform refinement property; von Neumann regular rings

Domaines: Mathématiques générales [math.GM]

Friedrich Wehrung. A uniform refinement property for congruence lattices. Proceedings of the American Mathematical Society, 1999, 127 (2), pp.363-370. ⟨10.1090/S0002-9939-99-04558-X⟩. ⟨hal-00004063⟩