partition lattice

The partition lattice (or lattice of partitions) ${\mathrm{\Pi }}_n$ is the lattice of set partitions (http://planetmath.org/Partition) of the set $[n]=\{1,\mathrm{\dots },n\}$. The partial order on ${\mathrm{\Pi }}_n$ is defined by refinement, setting $x\le y$ if any only if each cell of $x$ is contained in a cell of $y$.

If , then ${\mathrm{\Pi }}_n$ is a chain. But ${\mathrm{\Pi }}_3$ is not even a distributive lattice: