Order type

If a set ${\displaystyle X}$ has order type denoted ${\displaystyle \sigma }$, the order type of the reversed order, the dual of ${\displaystyle X}$, is denoted ${\displaystyle \sigma ^{*}}$.

The order type of a well-ordered set X is sometimes expressed as ord(X).[1]

The order type of the integers and rationals is usually denoted ${\displaystyle \pi }$ and ${\displaystyle \eta }$, respectively. The set of integers and the set of even integers have the same order type, because the mapping ${\displaystyle n\mapsto 2n}$ is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them. The open interval (0, 1) of rationals is order isomorphic to the rationals, since, for example, ${\displaystyle f(x)={\tfrac {2x-1}{1-\vert {2x-1}\vert }}}$ is a strictly increasing bijection from the former to the latter. Relevant theorems of this sort are expanded upon below.

More examples can be given now: The set of positive integers (which has a least element), and that of negative integers (which has a greatest element). The natural numbers have order type denoted by ω, as explained below.

The rationals contained in the half-closed intervals [0,1) and (0,1], and the closed interval [0,1], are three additional order type examples.

Three well-orderings on the set of natural numbers with distinct order types (top to bottom): ${\displaystyle \omega }$, ${\displaystyle \omega +5}$, and ${\displaystyle \omega +\omega }$.

Every well-ordered set is order-equivalent to exactly one ordinal number, by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. Order types thus often take the form of arithmetic expressions of ordinals.

With respect to their standard ordering as numbers, the set of rationals is not well-ordered. Neither is the completed set of reals, for that matter.

Any countable totally ordered set can be mapped injectively into the rational numbers in an order-preserving way. When the order is moreover dense and has no highest nor lowest element, there even exist a bijective such mapping.

• Well-order

• Weisstein, Eric W. "Order Type". MathWorld.