Can the distributive lattice theorem be improved?

Question

The theorem I am referring to is the following: A lattice $L$ is distributive iff none of its sublattices is isomorphic to $M_3$ or $N_5$. Can this theorem be improved? More precisely, is there a single lattice $X$ such that a lattice $L$ is distributive iff none of its sublattices is isomorphic to $X$?

Answer 1

No, as $M_3$ does not contain $N_5$ or vice versa, we need both in such a theorem.

Note the simularity to the planarity of graphs theorem by Kuratowski where we also have two "forbidden sub-objects" to characterise planarity. There a similar reasoning applies why we need two.