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 $\RR$:
- $\forall P, Q \in S / {\sim_\RR}: \exists p \in P, q \in Q: p \mathrel \RR q$
This article is complete as far as it goes, but it could do with expansion. In particular: The above is a second definition of Definition:Ordering Induced by Preordering, which we might do well to document. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Then $\struct {S / {\sim_\RR}, \preccurlyeq}$ is the antisymmetric quotient of $\struct {S, \RR}$.