A solution to Dilworth's Congruence Lattice Problem

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: Lattice; algebraic; distributive; congruence; semilattice; weakly distributive; free set

Domaines: Anneaux et algèbres [math.RA]; Mathématiques générales [math.GM]

Friedrich Wehrung. A solution to Dilworth's Congruence Lattice Problem. Advances in Mathematics, 2007, 216 (2), pp.610--625. ⟨10.1016/j.aim.2007.05.016⟩. ⟨hal-00016422v5⟩