Restriction of Ordering is Ordering

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 $T$.

Also see

 * Restriction of Equivalence Relation is Equivalence


 * Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.