Definition:Order Isomorphism/Definition 3

Definition
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: S \to T$ be a bijection such that:
 * $\forall x, y \in S: x \preceq_1 y \iff \map \phi x \preceq_2 \map \phi y$

Then $\phi$ is an order isomorphism.

Also see

 * Equivalence of Definitions of Order Isomorphism