\begin{definition}[Definition:Order Isomorphism/Definition 1]
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a bijection such that:
:$\phi: S \to T$ is order-preserving
:$\phi^{-1}: T \to S$ is order-preserving.
Then $\phi$ is an '''order isomorphism'''.
That is, $\phi$ is an '''order isomorphism''' {{iff}}:
:$\phi$ is bijective
:$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$
:$\forall p, q \in T: p \preceq_2 q \implies \map {\phi^{-1} } p \preceq_1 \map {\phi^{-1} } q$
So an '''order isomorphism''' can be described as a bijection that preserves ordering '''in both directions'''.
\end{definition}