\begin{definition}[Definition:Order Isomorphism/Definition 2]
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a surjective order embedding.
Then $\phi$ is an '''order isomorphism'''.
That is, $\phi$ is an '''order isomorphism''' {{iff}}:
:$(1): \quad \phi$ is surjective
:$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$
\end{definition}