In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) such that both f and its inverse are monotonic (preserving orders of elements).
In the special case when X is totally ordered, monotonicity of f already implies monotonicity of its inverse.
One and the same set may be equipped with different orders. Since orderequivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.
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Transcription
Notation
If a set has order type denoted , the order type of the reversed order, the dual of , is denoted .
The order type of a wellordered set X is sometimes expressed as ord(X).^{[1]}
Examples
The order type of the integers and rationals is usually denoted and , respectively. The set of integers and the set of even integers have the same order type, because the mapping 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 orderpreserving bijective mapping between them. The open interval (0, 1) of rationals is order isomorphic to the rationals, since, for example, 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 halfclosed intervals [0,1) and (0,1], and the closed interval [0,1], are three additional order type examples.
Order type of wellorderings
Every wellordered set is orderequivalent 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 wellordered set is usually identified with the corresponding ordinal. Order types thus often take the form of arithmetic expressions of ordinals.
Examples
Firstly, the order type of the set of natural numbers is ω. Any other model of Peano arithmetic, that is any nonstandard model, starts with a segment isomorphic to ω but then adds extra numbers. For example, any countable such model has order type ω + (ω* + ω) ⋅ η.
Secondly, consider the set V of even ordinals less than ω ⋅ 2 + 7:
As this comprises two separate counting sequences followed by four elements at the end, the order type is
Rational numbers
With respect to their standard ordering as numbers, the set of rationals is not wellordered. 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 orderpreserving way. When the order is moreover dense and has no highest nor lowest element, there even exist a bijective such mapping.
See also
External links
References
 ^ "Ordinal Numbers and Their Arithmetic". Archived from the original on 20091027. Retrieved 20070613.