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 $T$.

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 $T$.

$\blacksquare$

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

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 $1$