Example 4

In the set $\mathfrak{P}(\mathfrak{P}(\mathrm{E}))$ of sets of subsets of a set E, let $\mathscr{P}$ be the set of partitions of E (Chapter II, § 4, no. 7). We recall that a partition $\varpi$ is said to be coarser than a partition $\varpi'$ if given any $\mathrm{Y} \in \varpi'$ there exists $\mathrm{X} \in \varpi$ such that $\mathrm{Y} \subset \mathrm{X}$ (Chapter II, § 4, no. 6). For each partition $\varpi$ of E let $\tilde{\varpi}$ be the graph of the equivalence relation defined by $\varpi$ on E (Chapter II, § 6, no. 2), that is to say, the union of the (mutually disjoint) sets $\mathrm{A} \times \mathrm{A}$, where A runs through $\varpi$. The relation "$\varpi$ is coarser than $\varpi'$" is immediately seen to be equivalent to $\tilde{\varpi} \supset \tilde{\varpi}'$, and is therefore an order relation on the set $\mathscr{P}$ between $\varpi$ and $\varpi'$.

An ordering on a set E is a correspondence $\Gamma = (\mathrm{G},\ \mathrm{E},\ \mathrm{E})$ with E as source and as target, and such that the relation $(x,\ y) \in \mathrm{G}$ is an order relation on E. By abuse of language we shall sometimes refer to the graph G of $\Gamma$ as an ordering on E. If $\mathrm{R}\}x,\ y\{$ is an order relation on E, it has a graph which is an ordering on E.