Definition:Order Isomorphism

This page is about isomorphisms in order theory. For other uses, see Definition:Isomorphism.

Definition

Definition 1

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ 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 $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ 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

$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$

Two ordered sets $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are (order) isomorphic if there exists such an order isomorphism between them.

$\left({S, \preceq_1}\right)$ is described as (order) isomorphic to (or with) $\left({T, \preceq_2}\right)$, and vice versa.

This may be written $\left({S, \preceq_1}\right) \cong \left({T, \preceq_2}\right)$.

Where no confusion is possible, it may be abbreviated to $S \cong T$.

Well-Ordered Sets

When $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ 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.

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.

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.): $\S \text I.2$