Definition:Ordering Induced by Preordering

Definition

Let $\struct {S, \RR}$ be a relational structure such that $\RR$ is a preordering.

Let $\sim_\RR$ denote the equivalence on $S$ induced by $\RR$:

$x \sim_\RR y$ if and only if $x \mathrel \RR y$ and $y \mathrel \RR x$

Let a relation $\preccurlyeq_\RR$ be defined on the quotient set $S / {\sim_\RR}$ by:

$\eqclass x {\sim_\RR} \preccurlyeq_\RR \eqclass y {\sim_\RR} \iff x \mathrel \RR y$

where $\eqclass x {\sim_\RR}$ denotes the equivalence class of $x$ under $\sim_\RR$.

Then $\preccurlyeq_\RR$ is known as the ordering induced by $\RR$.

Also see

• Preordering induces Ordering for a proof that $\sim_\RR$ is indeed an ordering.

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations