# Embedding finite lattices into the lattice of partitions of a finite set

For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$.

Let $L$ be a finite lattice. Is there a finite set such that there is an injective lattice homomorphism $\varphi:L\to\text{Part}(S)$ for some finite set $S$?

• Whitman's result does not guarantee that $S$ is finite, which makes the problem much easier. – Eric Wofsey Jul 7 '15 at 10:57