\begin{definition}[Definition:Order Isomorphism/Wosets]
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be well-ordered sets.
Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving:
:$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$
Then $\phi$ is an '''order isomorphism'''.
Two well-ordered sets $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are '''(order) isomorphic''' if there exists such an '''order isomorphism''' between them.
Thus $\struct {S, \preceq_1}$ is described as '''(order) isomorphic to''' (or '''with''') $\struct {T, \preceq_2}$, and vice versa.
This may be written $\struct {S, \preceq_1} \cong \struct {T, \preceq_2}$.
Where no confusion is possible, it may be abbreviated to $S \cong T$.
\end{definition}