On the length of the congruence lattice of a lattice (Q5905374)

The author verifies the conjecture of E. T. Schmidt about the existence of a finite lattice of length at most \(5n\) whose congruence lattice is isomorphic to a given distributive lattice with \(n\) dual atoms. Namely, the author proves two main theorems: 1) Let \(D\) be a finite distributive lattice having \(n\) dual atoms. Then there is a finite lattice \(L\) of length \(5n\) such that the congruence lattice of \(L\) is isomorphic to \(D\). 2) For any integer \(n\), there exists a finite distributive lattice \(D_ n\) such that \(D_ n\) has \(n\) dual atoms and any lattice \(L\) whose congruence lattice is isomorphic to \(D_ n\) has length at least \(5n\). (To prove these theorems, the author uses the fact that a finite distributive lattice \(D\) has \(n\) dual atoms if and only if the poset of all join-irreducibles of \(D\) has \(n\) maximal elements).