# 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

- 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.

- 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**.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.28$ - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.7$: Well-Orderings and Ordinals