# Definition:Order Isomorphism

*This page is about Isomorphism in the context of Order Theory. For other uses, see Isomorphism.*

## Definition

### Definition 1

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

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

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

Then $\phi$ is an **order isomorphism**.

That is, $\phi$ is an **order isomorphism** if and only if:

- $(1): \quad \phi$ is surjective
- $(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

### Well-Ordered Sets

When $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ are well-ordered sets, 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 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

- Equivalence of Definitions of Order Isomorphism
- Definition:Relation Isomorphism, from which it can be seen that order isomorphism is a special case.
- Inverse of Increasing Bijection need not be Increasing

- 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**.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.28$

- 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S \text I.2$