Definition:Order Embedding/Definition 4

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

An order embedding is a mapping $\phi: S \to T$ such that the restriction of $\phi$ to $S \times \phi \left({S}\right)$ is an order isomorphism between $\left({S, \preceq_1}\right)$ and $\left({\phi \left({S}\right),\, {\preceq_2}{\restriction_{\phi \left({S}\right) \times \phi \left({S}\right)}} }\right)$.