Definition:Order Isomorphism/Definition 1

Definition

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.

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

$\phi$ is bijective

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

$\forall p, q \in T: p \preceq_2 q \implies \phi^{-1} \left({p}\right) \preceq_1 \phi^{-1} \left({q}\right)$

So an order isomorphism can be described as a bijection that preserves ordering in both directions.

Also see

• Equivalence of Definitions of Order Isomorphism

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

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 14$
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.2$: Order-preserving mappings. Isomorphisms
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations