\begin{definition}[Definition:Ordering Induced by Preordering]
Let $\struct {S, \RR}$ be a relational structure such that $\RR$ is a preordering.
Let $\sim_\RR$ denote the equivalence on $S$ induced by $\RR$:
:$x \sim_\RR y$ {{iff}} $x \mathrel \RR y$ and $y \mathrel \RR x$
Let a relation $\preccurlyeq_\RR$ be defined on the quotient set $S / {\sim_\RR}$ by:
:$\eqclass x {\sim_\RR} \preccurlyeq_\RR \eqclass y {\sim_\RR} \iff x \mathrel \RR y$
where $\eqclass x {\sim_\RR}$ denotes the equivalence class of $x$ under $\sim_\RR$.
Then $\preccurlyeq_\RR$ is known as the '''ordering induced by $\RR$'''.
\end{definition}