Definition:Antisymmetric Quotient

Definition
Let $\left({S, \precsim}\right)$ be a preordered set.

Let $\sim$ be the equivalence relation on $S$ induced by $\precsim$.

Let $S / {\sim}$ be the quotient set of $S$ by $\sim$.

Let $\preceq$ be the relation on $S / {\sim}$ defined by letting $P \preceq Q$ iff:


 * $\exists p \in P: \exists q \in Q: p \precsim q$

Then $\left({S / {\sim}, \preceq}\right)$ is the antisymmetric quotient of $\left({S, \precsim}\right)$.

Also see

 * Antisymmetric Quotient of Preordered Set is Ordered Set