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| In [[order theory|mathematical order theory]], an '''order-embedding''' is a special kind of [[monotone function]], which provides a way to include one [[partially ordered set]] into another. Like [[Galois connection]]s, order-embeddings constitute a notion which is strictly weaker than the concept of an [[order isomorphism]]. Both of these weakenings may be understood in terms of [[category theory]].
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| == Formal definition ==
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| Formally, given two partially ordered sets (''S'', ≤) and (''T'', ≤), a function ''f'': ''S'' → ''T'' is an ''order-embedding'' if ''f'' is both [[order-preserving]] and [[order-reflecting]], i.e. for all ''x'' and ''y'' in ''S'', one has
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| : <math>x\leq y \text{ if and only if } f(x)\leq f(y).</math><ref name="dp02">{{citation
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| | last1 = Davey | first1 = B. A.
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| | last2 = Priestley | first2 = H. A.
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| | contribution = Maps between ordered sets
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| | edition = 2nd
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| | isbn = 0-521-78451-4
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| | location = New York
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| | mr = 1902334
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| | pages = 23–24
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| | publisher = Cambridge University Press
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| | title = Introduction to Lattices and Order
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| | url = http://books.google.com/books?id=vVVTxeuiyvQC&pg=PA23
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| | year = 2002}}.</ref>
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| Note that such a function is necessarily [[injective]], since ''f''(''x'') = ''f''(''y'') implies ''x'' ≤ ''y'' and ''y'' ≤ ''x''.<ref name="dp02"/> If an order-embedding between two posets ''S'' and ''T'' exists, one says that ''S'' can be embedded into ''T''.
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| == Properties ==
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| An order isomorphism can be characterized as a [[surjective]] order-embedding. As a consequence, any order-embedding ''f'' restricts to an isomorphism between its [[domain (mathematics)|domain]] ''S'' and its [[range (mathematics)|range]] ''f''(''S''), which justifies the term "embedding".<ref name="dp02"/> On the other hand, it might well be that two (necessarily infinite) posets are mutually embeddable into each other without being isomorphic. An example is provided by the set of [[real number]]s and its [[interval (mathematics)|interval]] [−1,1]. Ordering both sets in the natural way, one clearly finds that [−1,1] can be embedded into the reals. On the other hand, one can define a function ''e'' from the real numbers to [−1,1] as<ref>For a very similar example using arctan to order-embed the real numbers into an interval, see {{citation|title=Discovering Modern Set Theory: The basics|volume=8|series=Fields Institute Monographs|first1=Winfried|last1=Just|first2=Martin|last2=Weese|publisher=American Mathematical Society|year=1996|isbn=9780821872475|page=21|url=http://books.google.com/books?id=TPvHr7fcvHoC&pg=PA21}}.</ref>
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| : <math>e(x) = \frac{2}{\pi}\arctan x</math>
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| This is the projection of the real number line to (half of) the circle with circumference 4 (see [[trigonometric function]]s for details) and embeds the reals into [−1,1]. Yet, the two posets are not isomorphic: [−1,1] has both a [[least element|least]] and a [[greatest element]], which are not present in the case of the real numbers. This shows that an isomorphism cannot exist.
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| In a retract (a pair of order-preserving maps whose composition is the identity), the first of the two maps (called a coretraction) must be an order-embedding.<ref>{{citation
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| | last1 = Duffus | first1 = Dwight
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| | last2 = Laflamme | first2 = Claude
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| | last3 = Pouzet | first3 = Maurice
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| | arxiv = math/0612458
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| | doi = 10.1007/s00012-008-2125-6
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| | issue = 1-2
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| | journal = Algebra Universalis
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| | mr = 2453498
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| | pages = 243–255
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| | title = Retracts of posets: the chain-gap property and the selection property are independent
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| | volume = 59
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| | year = 2008}}.</ref> However, not every order-embedding is a coretraction. As a trivial example of this phenomenon, the unique order embedding from the empty poset to a nonempty poset ''P'' has no retract, because there is no order-preserving map from ''P'' to the empty poset. More illustratively, consider the "diamond poset" with elements {00, 01, 10, 11} with 00 < 01 < 11, 00 < 10 < 11 and 01 incomparable to 10. Consider the embedded sub-poset "V" consisting of {00, 01, 10} (with 00 < 01 and 00 < 10 and 01 incomparable to 10). A retract of the embedding V -> diamond would need to send 11 to somewhere in "V" above both 01 and 10, but there is no such place.
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| == Additional Perspectives ==
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| {{unreferenced section|date=October 2013}}
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| Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example:
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| * (Model theoretically) A poset is a set equipped with a (reflexive, antisymmetric, transitive) binary relation. An order embedding A -> B is an isomorphism from A to an [[elementary substructure]] of B.
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| * (Graph theoretically) A poset is a (transitive, acyclic, directed, reflexive) graph. An order embedding A -> B is a [[graph isomorphism]] from A to an [[induced subgraph]] of B.
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| * (Category theoretically) A poset is a (small, skeletal) category such that each homset has at most one element. An order embedding A -> B is a full and faithful functor from A to B which is injective on objects, or equivalently an isomorphism from A to a [[full subcategory]] of B.
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| ==References==
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| {{reflist}}
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| [[Category:Order theory]]
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