Let $\mathcal{C}$ be a collection of subsets of $[n]$ that is closed under taking unions (including taking the union of the empty set which is the empty set). Then $(\mathcal{C},\subseteq)$ is a lattice. But is the converse true? In other words, given any lattice is there a union closed family of subsets of $[n]$ (for some $n$) whose members obey the order of the lattice?
I looked at all the lattices with up to 6 points and haven't found a counter-example but I can't come up with any algorithm that would prove the statement.
P.S. I do know that the converse is true for distributive lattices. Also, I am only considering finite lattices.