Lattice of partitions

Example

Let $n \ge 1$ and let $\Pi_n$ be the set of all partitions of $\set{1, \ldots, n}$. Order $\Pi_n$ by refinement: $\sigma \le \tau$ when each block of $\sigma$ lies inside some block of $\tau$. This is a partial order, and $\Pi_n$ is a lattice. The meet $\sigma \wedge \tau$ has as blocks the nonempty intersections $B \cap C$ of a block $B$ of $\sigma$ and a block $C$ of $\tau$. The join $\sigma \vee \tau$ is the finest partition coarser than both. In it, two points share a block when a chain of blocks of $\sigma$ and $\tau$, each meeting the next, connects them. The least element is the partition into singletons, and the greatest element is the partition with one block. The number of elements of $\Pi_n$ is the Bell number $B_n$.

Check for $n = 3$. The five partitions are $1|2|3$, $12|3$, $13|2$, $23|1$ and $123$. The three partitions with one pair lie strictly between the bottom and the top, and no two of them are comparable. The meet of any two of them is $1|2|3$ and their join is $123$, so $\Pi_3$ is the diamond lattice $M_3$.