A solution to Dilworth's Congruence Lattice Problem
Résumé
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. If we restrict our attention to lattices with m-permutable congruences, then we may take D with aleph 2^m compact elements. Our results extend to all algebras possessing a polynomially definable structure of bounded semilattice.