Restriction of Ordering is Ordering
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Theorem
Let $S$ be a set.
Let $\preceq$ be an ordering on $S$.
Let $T \subseteq S$ be a subset of $S$.
Let $\preceq \restriction_T$ be the restriction of $\preceq$ to $T$.
Then $\preceq \restriction_T$ is an ordering on $T$.
Proof
Let $\preceq$ be an ordering on $S$.
Then, by definition:
- $\preceq$ is a reflexive relation on $S$
- $\preceq$ is an antisymmetric relation on $S$
- $\preceq$ is a transitive relation on $S$.
Then:
- from Restriction of Reflexive Relation is Reflexive, $\preceq \restriction_T$ is a reflexive relation on $T$
- from Restriction of Antisymmetric Relation is Antisymmetric, $\preceq \restriction_T$ is an antisymmetric relation on $T$
- from Restriction of Transitive Relation is Transitive, $\preceq \restriction_T$ is a transitive relation on $T$
and so it follows by definition that $\preceq \restriction_T$ is an ordering on $T$.
$\blacksquare$
Also see
- Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.1$: Theorem $1$