Definition:Antisymmetric Quotient

Definition
Let $\struct {S, \precsim}$ 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 $P \preceq Q$ :


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

Then $\struct {S / {\sim}, \preceq}$ is the antisymmetric quotient of $\struct {S, \precsim}$.

Also see

 * Antisymmetric Quotient of Preordered Set is Ordered Set