Exercise 18, § 1

Let A be a set with at least three elements, let $\mathfrak{P}$ be the set of all partitions of A, ordered by the relation "$\varpi$ is finer than $\varpi'$" between $\varpi$ and $\varpi'$ (no. 1, Example 4). Show that $\mathfrak{P}$ is a complete lattice (Exercise 11), is not distributive (Exercise 17), but is relatively complemented. (To prove the last assertion, well-order the sets belonging to a partition.) \*