Editing Order embedding (section)

== Properties ==
[[File:Mutual embedding of open and closed real unit interval svg.svg|thumb|300px|Mutual order embedding of <math>(0,1)</math> and <math>[0,1]</math>, using <math>f(x) = (94x+3)/100</math> in both directions.]]
[[File:Lattice T(6).svg|thumb|The set <math>S</math> of divisors of 6, partially ordered by ''x'' divides ''y''. The embedding <math>id: \{ 1,2,3 \} \to S</math> cannot be a coretraction.]]
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 of a function|domain]] ''S'' and its [[image (mathematics)|image]] ''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 order-embeddable into each other without being order-isomorphic. 

An example is provided by the [[open interval]] <math>(0,1)</math> of [[real number]]s and the corresponding [[closed interval]] <math>[0,1]</math>. The function <math>f(x) = (94x+3) / 100</math> maps the former to the [[subset]] <math>(0.03,0.97)</math> of the latter and the latter to the subset <math>[0.03,0.97]</math> of the former, see picture. Ordering both sets in the natural way, <math>f</math> is both order-preserving and order-reflecting (because it is an [[linear function|affine function]]<!---[[affine function]] talks only about geometry--->). Yet, no isomorphism between the two posets can exist, since e.g. <math>[0,1]</math> has a [[least element]] while <math>(0,1)</math> does not.
For a similar example using arctan to order-embed the real numbers into an interval, and the [[identity map]] for the reverse direction, see e.g. Just and Weese (1996).<ref>{{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=https://books.google.com/books?id=TPvHr7fcvHoC&pg=PA21}}</ref>

A retract is a pair <math>(f,g)</math> of order-preserving maps whose [[function composition|composition]] <math>g \circ f</math> is the identity. In this case, <math>f</math> is called a coretraction, and must be an order embedding.<ref>{{citation
 | last1 = Duffus | first1 = Dwight
 | last2 = Laflamme | first2 = Claude
 | last3 = Pouzet | first3 = Maurice
 | arxiv = math/0612458
 | doi = 10.1007/s00012-008-2125-6
 | issue = 1–2
 | journal = Algebra Universalis
 | mr = 2453498
 | pages = 243–255
 | title = Retracts of posets: the chain-gap property and the selection property are independent
 | volume = 59
 | year = 2008| s2cid = 14259820
 }}.</ref> However, not every order embedding is a coretraction. As a trivial example, the unique order embedding <math>f: \emptyset \to \{1\}</math> from the empty poset to a nonempty poset has no retract, because there is no order-preserving map <math>g: \{1\} \to \emptyset</math>. More illustratively, consider the set <math>S</math> of [[divisor]]s of 6, partially ordered by ''x'' [[divides]] ''y'', see picture. Consider the embedded sub-poset <math>\{ 1,2,3 \}</math>.  A retract of the embedding <math>id: \{ 1,2,3 \} \to S</math> would need to send <math>6</math> to somewhere in <math>\{ 1,2,3 \}</math> above both <math>2</math> and <math>3</math>, but there is no such place.