# Definition:Order Isomorphism

## Definition

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

### Definition 1

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.

### Definition 2

Let $\phi: S \to T$ be a surjective order embedding.

Then $\phi$ is an order isomorphism.

### Definition 3

Let $\phi: S \to T$ be a bijection such that:

$\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

Then $\phi$ is an order isomorphism.

### Well-Orderings

When $\preceq_1$ and $\preceq_2$ are well-orderings, the condition on the order preservation can be relaxed:

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.

## Isomorphic Sets

Two 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.

Hence $\struct {S, \preceq_1}$ 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 likely to arise, it can be abbreviated to $S \cong T$.

## Examples

### Real Arctangent Function

The real arctangent function $\arctan$ is an order isomorphism between the set of real numbers $\R$ and the open real interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ under the usual ordering.

## Also see

• Results about order isomorphisms can be found here.

## Historical Note

The concept of order isomorphism was first introduced by Georg Cantor.

## 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.