Definition:Antisymmetric Quotient

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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