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 iff:


 * $\phi$ is surjective and
 * $\forall x, y \in S: x \mathop {\preceq_1} y \iff \phi \left({x}\right) \mathop {\preceq_2} \phi \left({y}\right)$

Also see

 * Equivalence of Definitions of Order Isomorphism