Definition:Order Embedding/Definition 4

Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

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

Let $T' = \operatorname{Im} \left({S}\right)$ 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 $\left({S, \preceq_1}\right)$ and $\left({T', \preceq_2 \restriction_{T' \times T'} }\right)$.

Also see

 * Equivalence of Definitions of Order Embedding