# 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 if and only 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 $S \cong T$.

### Class-Theoretical Definition

In the context of class theory, the definition follows the same lines:

Let $\struct {A, \preccurlyeq_1}$ and $\struct {B, \preccurlyeq_2}$ be well-ordered classes.

Let $\phi: A \to B$ be a bijection such that $\phi: A \to B$ is order-preserving:

$\forall x, y \in S: x \preccurlyeq_1 y \implies \map \phi x \preccurlyeq_2 \map \phi y$

Then $\phi$ is an order isomorphism.

## Also see

• Results about order isomorphisms can be found here.

## Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.