Definition:Order Isomorphism/Definition 2

Definition

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

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.