Definition:Order Isomorphism/Definition 2

Definition
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ 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 :


 * $(1): \quad \phi$ is surjective
 * $(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

Also see

 * Equivalence of Definitions of Order Isomorphism