TheIn [[mathematics]], the '''Congruencecongruence Latticelattice Problemproblem''' asks whether every [[Distributivity (order theory)|distributive]] [[compact element|algebraic lattice]] is [[isomorphic]] to the congruence lattice of (some other) lattice. ItFor many years it was one of the most famous and long-standing open problems in [[Lattice (order)|lattice theory]], anand areahad ofa [[mathematics]].deep Besidesimpact beingit openhad foron athe longdevelopment time,of itlattice wastheory distinguisheditself. byThe variousconjecture relatedis problemstrue infor otherall areasdistributive oflattices mathematicswith andat bymost a[[Aleph profoundnumber|ℵ<sub>1</sub>]] impact[[compact itelement]]s, hadbut onF. theWehrung developmentprovided ofa latticecounterexample theoryfor itself.distributive Thelattices problemwith wasℵ<sub>2</sub> recentlycompact solvedelements by F. Wehrung, inusing a mostconstruction surprisingbased fashion,on by[[Kuratowski's afree counterexampleset constructiontheorem]].
The surprising aspect of the construction is twofold. There exists a counterexample with ℵ<sub>2</sub> many compact elements, though all distributive algebraic lattices with at most ℵ<sub>1</sub> many compact elements satisfy the conjecture. Also, this was proved without any additional assumptions to the standard ZFC axiom system of [[set theory]]. The key element which distinguishes ℵ<sub>2</sub> from ℵ<sub>1</sub> is the almost-forgotten old infinite combinatorics result by [[Kazimierz Kuratowski|Kuratowski]] called the [[Kuratowski's Free Set Theorem|Free Set Theorem]].
