Definition:Order Isomorphism

Definition 2
Two ordered sets $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are (order) isomorphic if there exists such an order isomorphism between them.

$\left({S, \preceq_1}\right)$ is described as (order) isomorphic to (or with) $\left({T, \preceq_2}\right)$, and vice versa.

This may be written $\left({S, \preceq_1}\right) \cong \left({T, \preceq_2}\right)$.

Where no confusion is possible, it may be abbreviated to $S \cong T$.

Well-Ordered Sets
When $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ are well-ordered sets, the condition on the order preservation can be relaxed:

Also see

 * Equivalence of Definitions of Order Isomorphism
 * Definition:Relation Isomorphism, from which it can be seen that order isomorphism is a special case.
 * Inverse of Increasing Bijection need not be Increasing


 * Definition:Order Embedding