\begin{definition}[Definition:Antisymmetric Quotient]
Let $\struct {S, \RR}$ be a preordered set.
Let $\sim_\RR$ be the equivalence relation on $S$ induced by $\RR$.
Let $S / {\sim_\RR}$ be the quotient set of $S$ by $\sim_\RR$.
Let $\preccurlyeq$ be the ordering on $S / {\sim_\RR}$ induced by $\RR$:
:$\forall P, Q \in S / {\sim_\RR}: \exists p \in P, q \in Q: p \mathrel \RR q$
{{expand|The above is a second definition of Definition:Ordering Induced by Preordering, which we might do well to document.}}
Then $\struct {S / {\sim_\RR}, \preccurlyeq}$ is the '''antisymmetric quotient''' of $\struct {S, \RR}$.
\end{definition}