Definition:Order Isomorphism/Well-Orderings

Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be well-ordered sets.

Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving:
 * $\forall x, y \in S: x \mathop {\preceq_1} y \implies \phi \left({x}\right) \mathop {\preceq_2} \phi \left({y}\right)$

Then $\phi$ is an order isomorphism.

Two well-ordered sets $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are (order) isomorphic if there exists such an order isomorphism between them.

Thus $\left({S, \preceq_1}\right)$ is described as (order) isomorphic to (or with) $\left({T, \preceq_2}\right)$, and vice versa.

This may be written $\left({S, \preceq_1}\right) \cong \left({T, \preceq_2}\right)$.

Where no confusion is possible, it may be abbreviated to $S \cong T$.

Also see

 * Order-Preserving Bijection on Wosets is Order Isomorphism, where it is shown that this definition is compatible with that of an order isomorphism between ordered sets.