Definition:Order Isomorphism

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

• 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

• Definition:Order Embedding

• 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

This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code.

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