\begin{definition}[Definition:Preordering/Definition 2]
Let $\RR \subseteq S \times S$ be a relation on a set $S$.
$\RR$ is a '''preordering''' on $S$ {{iff}}:
:$(1): \quad \RR \circ \RR = \RR$
:$(2): \quad \Delta_S \subseteq \RR$
where:
:$\circ$ denotes relation composition
:$\Delta_S$ denotes the diagonal relation on $S$.
\end{definition}