\begin{definition}[Definition:Order Embedding/Definition 1]
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}}:
:$\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$
That is, an '''order embedding''' is an order-preserving, order-reflecting mapping.
\end{definition}