Example 2

In particular, $\emptyset$ is the least element and E the greatest element of $\mathfrak{P}(E)$. In the set $\Phi(E, F)$ of mappings of subsets of E into F, ordered by extension of mappings (no. 1, Example 3), the empty mapping is the least element, and there is no greatest element unless F consists of a single element. The diagonal $\Delta$ of $E \times E$ is the least element of the set of graphs of equivalence relations on E (or of the set of preorderings on E).