Let $L$ be a lattice. Is there a union closed family of subsets of $[n]$ ordered by inclusion that correspond to $L$.

Question

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.

Answer 1

The converse is true.

If $L$ is a lattice and $x\in L$, define $A_x = \{z\in L\;|\;z\not\geq x\}$. The function $A\colon L\to {\mathcal P}(L)\colon x\mapsto A_x$ is a join-embedding of $(L,\vee)$ into $({\mathcal P}(L),\cup)$. The collection $\{A_x\;|\;x\in L\}$ is therefore a union-closed set of subsets of $L$ which, when ordered by inclusion, forms a lattice isomorphic to $L$.