Definition:Order Embedding/Definition 1

Definition

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: S \to T$ be a mapping.

$\phi$ is an order embedding of $S$ into $T$ if and only if:

$\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

That is, an order embedding is an order-preserving, order-reflecting mapping.

Also defined as

It is usual to state in the definition for an order embedding that it be injective.

As can be seen in Order Embedding is Injection, that condition is redundant.

Also known as

An order embedding is also known as an order monomorphism.

Some sources call it an order-preserving mapping, but this term is also used (in particular on $\mathsf{Pr} \infty \mathsf{fWiki}$ to be the same thing as an increasing mapping: that is, a mapping which preserves an ordering in perhaps only one direction.

Also see

• Equivalence of Definitions of Order Embedding

• Results about order embeddings can be found here.

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: Definition $5$