Definition:Order Isomorphism

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

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

This may be written $\struct {S, \preceq_1} \cong \struct {T, \preceq_2}$.

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

Well-Ordered Sets
When $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ 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