Definition:Antisymmetric Quotient

Definition
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 $\RR$:


 * $\forall P, Q \in S / {\sim_\RR}: \exists p \in P, q \in Q: p \mathrel \RR q$

Then $\struct {S / {\sim_\RR}, \preccurlyeq}$ is the antisymmetric quotient of $\struct {S, \RR}$.

Also see

 * Antisymmetric Quotient of Preordered Set is Ordered Set