A solution to Dilworth's Congruence Lattice Problem
Rings and Algebras
2007-11-10 v5
Abstract
We construct a distributive algebraic lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R. P. Dilworth, from the forties. The lattice D has compact top element and aleph omega+1 compact elements. Our results extend to all algebras possessing a polynomially definable structure of a join-semilattice with a largest element.
Cite
@article{arxiv.math/0601059,
title = {A solution to Dilworth's Congruence Lattice Problem},
author = {Friedrich Wehrung},
journal= {arXiv preprint arXiv:math/0601059},
year = {2007}
}
Comments
Version 1 presents a longer and slightly more general proof, based on so-called "uniform refinement properties". Version 2 presents a shorter proof. Versions 3 an 4 add a few minor improvements. Version 5 fixes a minor oversight in the proof of Theorem 7.1