\begin{definition}[Definition:Order Embedding/Definition 2]
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 $T$''' {{iff}} both of the following conditions hold:
:$(1): \quad \phi$ is an injection
:$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$
\end{definition}