Proof of 1-dimensional congruence lattice problem for distributive lattices
by
Friedrich Wehrung
CNRS, ESA 6081, Université de Caen, département de Mathématiques, 14032 Caen cedex.

We outline a proof of the following result (full version in preprint ``Forcing extensions of partial lattices'', available at URL indicated below):

Let K be a lattice, let D be a distributive lattice with zero, and let j: Con_c K --> D be a \/ , 0-homomorphism, where Con_c K denotes the \/ , 0-semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f: K --> L, and an isomorphism \alpha: Con_c L --> D such that \alpha o Con_c f=j.

Furthermore, L and f satisfy many additional properties, for example:

1. L is relatively complemented.
2. L has definable principal congruences.
3. If the range of j is cofinal in D, then the convex sublattice of L generated by f[K] equals L.

We mention the following corollaries, that extend many results obtained in the last decades in that area:

--- Every lattice K such that Con_c K is a lattice admits a congruence-preserving extension into a relatively complemented lattice.
--- Every \/ , 0-direct limit of a countable sequence of distributive lattices with zero is isomorphic to the semilattice of compact congruences of a relatively complemented lattice with zero.

http://www.math.unicaen.fr/~wehrung

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-81.