In [[mathematics]], the '''congruence lattice problem''' asks whether every [[distributive lattice]] is [[isomorphic]] to the congruence lattice of some other lattice. ForThe problem was posed by [[Robert P. Dilworth]], and for many years it was one of the most famous and long-standing open problems in [[lattice theory]],; andit had a deep impact it had on the development of lattice theory itself. The conjecture is true for all distributive lattices with at most [[Aleph number|ℵ<sub>1</sub>]] [[compact element]]s, but F. Wehrung provided a counterexample for distributive lattices with ℵ<sub>2</sub> compact elements using a construction based on [[Kuratowski's free set theorem]].
