# 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^{}: