Definitions

# 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: XY 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 $nmapsto2n$ preserves the order. But the set of integers and the set of rational numbers are not order isomorphic, because, even though the sets are the same size, 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 ($y = frac\left\{2x - 1\right\}\left\{1 - vert \left\{2x - 1\right\} vert\right\}$ provides a monotone bijection from the former to the latter), but the half-closed intervals [0,1) and (0,1], and the closed interval [0,1], supply three new examples for a total of seven order type examples in this paragraph.

Since order-equivalence is an equivalence relation, it partitions the class of all sets into equivalence classes. Every well-ordered set is order-equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is often identified with the corresponding ordinal. For example, the order type of the natural numbers is said to be ω.

The order type of a well-ordered set V is sometimes expressed as ord(V).

For example, consider the set of even ordinals less than ω·2+7, which is:

V = {0, 2, 4, 6, ...; ω, ω+2, ω+4, ...; ω·2, ω·2+2, ω·2+4, ω·2+6}.

Its order type is:

ord(V) = ω·2+4 = {0, 1, 2, 3, ...; ω, ω+1, ω+2, ...; ω·2, ω·2+1, ω·2+2, ω·2+3}.

## Notation

The order type of the rationals is usually denoted $eta$. If a set S has order type $sigma$, the order type of the dual of S (the reversed order) is denoted $sigma^\left\{*\right\}$.