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In the [[mathematics|mathematical]] field of [[order theory]] an '''order isomorphism''' is a special kind of [[monotone function]] that constitutes a suitable notion of [[isomorphism]] for [[partially ordered set]]s (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are [[order embedding]]s and [[Galois connection]]s.<ref>{{harvtxt|Block|2011}}; {{harvtxt|Ciesielski|1997}}.</ref>
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==Definition==
Formally, given two posets <math>(S,\le_S)</math> and <math>(T,\le_T)</math>, an '''order isomorphism'''  from <math>(S,\le_S)</math> to <math>(T,\le_T)</math> is a [[bijection|bijective function]] <math>f</math> from <math>S</math> to <math>T</math> with the property that, for every <math>x</math> and <math>y</math> in <math>S</math>, <math>x \le_S y</math> if and only if <math>f(x)\le_T f(y)</math>. That is, it is a bijective [[order-embedding]].<ref>This is the definition used by {{harvtxt|Ciesielski|1997}}. For {{harvtxt|Bloch|2011}} and {{harvtxt|Schröder|2003}} it is a consequence of a different definition.</ref>
 
It is also possible to define an order isomorphism to be a [[surjective]] order-embedding. The two assumptions that <math>f</math> cover all the elements of <math>T</math> and that it preserve orderings, are enough to ensure that <math>f</math> is also one-to-one, for if <math>f(x)=f(y)</math> then (by the assumption that <math>f</math> preserves the order) it would follow that <math>x\le y</math> and <math>y\le x</math>, implying by the definition of a partial order that <math>x=y</math>.
 
Yet another characterization of order isomorphisms is that they are exactly the [[monotone function|monotone]] [[bijection]]s that have a monotone inverse.<ref>This is the definition used by {{harvtxt|Bloch|2011}} and {{harvtxt|Schröder|2003}}.</ref>
 
An order isomorphism from a partially ordered set to itself is called an '''order [[automorphism]]'''.<ref>{{harvtxt|Schröder|2003}}, p. 13.</ref>
 
==Examples==
* The [[identity function]] on any partially ordered set is always an order automorphism.
* [[Additive inverse|Negation]] is an order isomorphism from <math>(\mathbb{R},\leq)</math> to <math>(\mathbb{R},\geq)</math> (where <math>\mathbb{R}</math> is the set of [[real number]]s and <math>\le</math> denotes the usual numerical comparison), since &minus;''x'' ≥ &minus;''y'' if and only if ''x'' ≤ ''y''.<ref>See example 4 of {{harvtxt|Ciesielski|1997}}, p. 39., for a similar example with integers in place of real numbers.</ref>
* The [[open interval]] <math>(0,1)</math> (again, ordered numerically) does not have an order isomorphism to or from the [[closed interval]] <math>[0,1]</math>: the closed interval has a least element, but the open interval does not, and order isomorphisms must preserve the existence of least elements.<ref>{{harvtxt|Ciesielski|1997}}, example 1, p. 39.</ref>
 
==Order types==
If <math>f</math> is an order isomorphism, then so is its [[inverse function]].
Also, if <math>f</math> is an order isomorphism from <math>(S,\le_S)</math> to <math>(T,\le_T)</math> and <math>g</math> is an order isomorphism from <math>(T,\le_T)</math> to <math>(U,\le_U)</math>, then the [[function composition]] of <math>f</math> and <math>g</math> is itself an order isomorphism, from <math>(S,\le_S)</math> to <math>(U,\le_U)</math>.<ref>{{harvtxt|Ciesielski|1997}}; {{harvtxt|Schröder|2003}}.</ref>
 
Two partially ordered sets are said to be '''order isomorphic''' when there exists an order isomorphism from one to the other.<ref>{{harvtxt|Ciesielski|1997}}.</ref> Identity functions, function inverses, and compositions of functions correspond, respectively, to the three defining characteristics of an [[equivalence relation]]: [[reflexive relation|reflexivity]], [[symmetric relation|symmetry]], and [[transitive relation|transitivity]]. Therefore, order isomorphism is an equivalence relation. The class of partially ordered sets can be partitioned by it into [[equivalence class]]es, families of partially ordered sets that are all isomorphic to each other. These equivalence classes are called [[order type]]s.
 
==See also==
*[[Permutation pattern]], a permutation that is order-isomorphic to a subsequence of another permutation
 
==Notes==
{{reflist|colwidth=30em}}
 
==References==
*{{citation
| last = Bloch | first = Ethan D.
| edition = 2nd
| isbn = 9781441971265
| pages = 276–277
| publisher = Springer
| series = Undergraduate Texts in Mathematics
| title = Proofs and Fundamentals: A First Course in Abstract Mathematics
| url = http://books.google.com/books?id=QJ_537n8zKYC&pg=PA276
| year = 2011}}.
*{{citation
| last = Ciesielski | first = Krzysztof
| isbn = 9780521594653
| pages = 38–39
| publisher = Cambridge University Press
| series = London Mathematical Society Student Texts
| title = Set Theory for the Working Mathematician
| url = http://books.google.com/books?id=tTEaMFvzhDAC&pg=PA38
| volume = 39
| year = 1997}}.
*{{citation
| last = Schröder | first = Bernd Siegfried Walter
| isbn = 9780817641283
| page = 11
| publisher = Springer
| title = Ordered Sets: An Introduction
| url = http://books.google.com/books?id=2esoXnolEWgC&pg=PA11&lpg=PA11
| year = 2003}}.
 
[[Category:Order theory]]
[[Category:Morphisms]]

Latest revision as of 17:21, 2 January 2015

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