Congruence lattice problem

Remarks

The question is natural because the congruence lattice of every lattice is distributive and algebraic. An equivalent form asks whether every distributive join-semilattice with a least element is isomorphic to the semilattice of compact lattice congruences of some lattice.

The answer is positive for finite lattices: every finite distributive lattice is isomorphic to the congruence lattice of a finite lattice. It is also positive when the semilattice of compact elements has at most $\aleph_1$ elements.

The problem was solved in the negative by Friedrich Wehrung, in a paper published in 2007. He built a counterexample whose semilattice of compact elements has $\aleph_{\omega+1}$ elements. Pavel Růžička later lowered the size of a counterexample to $\aleph_2$, which is optimal by the positive result above.