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

• 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

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

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