Definition:Order Isomorphism/Well-Orderings

Definition
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 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 $S \cong T$.

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.