Congruence lattices of lattices
The "Congruence Lattice Problem" in the Wikipedia
Problem 1. Does there exists a distributive algebraic lattice which is not isomorphic to the congruence lattice of any algebraic structure with only finitely many operations ? What lattices are congruence lattices of algebras with only finitely many operations?
Problem 2. ( Lampe) Is there an algebra A with exactly אi0 operations such that there is no B with only finitely many operations and with Con(B) isomorphic to Con(A) ?
Problem 3. Is every finite distributive lattice isomorphic to the congruence lattice of an isoform modular lattice ?
Problem 4. (Lampe) Which algebraic lattices are isomorphic to the congruence lattices of algebras having one element subalgebras ?