Proposition 1

A correspondence $\Gamma$ between E and E is an ordering on E if and only if its graph G satisfies the following conditions :

(a) $\mathrm{G} \circ \mathrm{G} = \mathrm{G}$.

(b) The set $\mathrm{G} \cap \overset{-1}{\mathrm{G}}$ is the diagonal $\Delta$ of $\mathrm{E} \times \mathrm{E}$.

For the relation $((x,\ y) \in \mathrm{G}$ and $(y,\ z) \in \mathrm{G}) \Rightarrow ((x,\ z) \in \mathrm{G})$ can be written as $\mathrm{G} \circ \mathrm{G} \subset \mathrm{G}$, and the relation

can be written as $\mathrm{G} \cap \overset{-1}{\mathrm{G}} = \Delta$. From $\mathrm{G} \cap \overset{-1}{\mathrm{G}} = \Delta$ we then deduce $\Delta \subset \mathrm{G}$, whence $\mathrm{G} = \Delta \circ \mathrm{G} \subset \mathrm{G} \circ \mathrm{G}$; since also $\mathrm{G} \circ \mathrm{G} \subset \mathrm{G}$, we have