Order Type

In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection f: X → Y such that both f and its inverse are monotone (order preserving). (In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse.)

For example, the set of integers and the set of even integers have the same order type, because the mapping preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) are not order isomorphic, because, even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positive integers (which has a least element), and that of negative integers (which has a greatest element). The open interval (0,1) of rationals is order isomorphic to the rationals (since

provides a monotone bijection from the former to the latter); the half-closed intervals, and the closed interval, are three additional order type examples.

Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.