To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

From Wikipedia, the free encyclopedia

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

YouTube Encyclopedic

  • 1/3
    13 606
    14 530
    5 514
  • Differential Equations: Definitions and Terminology (Level 1 of 4) | Order, Type, Linearity
  • Differential Equation - Introduction (12 of 15) Types 1, 2, 3 of First Order Differential Equations
  • Differential Equaitons: Find the Order and Classify as Linear or Nonlinear



If a set has order type denoted , the order type of the reversed order, the dual of , is denoted .

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

Order type of well-orderings

Three well-orderings on the set of natural numbers with distinct order types (top to bottom): , , and .

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.


Firstly, the order type of the set of natural numbers is ω. Any other model of Peano arithmetic, that is any non-standard 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 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.

See also

External links

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


  1. ^ "Ordinal Numbers and Their Arithmetic". Archived from the original on 2009-10-27. Retrieved 2007-06-13.
This page was last edited on 13 August 2023, at 19:20
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.