\begin{definition}[Definition:Ordering/Definition 2]
Let $S$ be a set.
An '''ordering on $S$''' is a relation $\RR$ on $S$ such that:
:$(1): \quad \RR \circ \RR = \RR$
:$(2): \quad \RR \cap \RR^{-1} = \Delta_S$
where:
:$\circ$ denotes relation composition
:$\RR^{-1}$ denotes the inverse of $\RR$
:$\Delta_S$ denotes the diagonal relation on $S$.
\end{definition}