Definition:Order Embedding/Definition 4

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

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

Let $T' = \Img S$ be the image of $S$ under $\phi$.

Then $\phi$ is an order embedding :
 * the restriction of $\phi$ to $S \times T'$ is an order isomorphism between $\struct {S, \preceq_1}$ and $\struct {T', \preceq_2 \restriction_{T' \times T'} }$.

Also see

 * Equivalence of Definitions of Order Embedding