Definition:Order Isomorphism/Definition 1

Definition

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.

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

$\phi$ is bijective

$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$

$\forall p, q \in T: p \preceq_2 q \implies \map {\phi^{-1} } p \preceq_1 \map {\phi^{-1} } q$

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

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.