Inclusion Mapping is Order Embedding

Theorem

Let $\struct {S, \preccurlyeq_S}$ and $\struct {T, \preccurlyeq_T}$ be ordered sets such that $S \subseteq T$.

Let ${\preccurlyeq_S} = {\preccurlyeq_T} {\restriction_S}$ be the restriction of $\preccurlyeq_T$ to $S$.

Let $i_S: S \to T$ denote the inclusion mapping from $S$ to $T$:

$\forall s \in S: \map {i_S} s = s$

Then $i_S$ is an order embedding.

Proof

We have that Inclusion Mapping is Restriction of Identity.

Then we have that Identity Mapping is Order Isomorphism.

The result follows.

$\blacksquare$

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