View source for Partial-function

{{Stub page|grade=D|msg=Flesh out, find some references, link to relations. This page was created mainly to make note of the partial version of a (total) function, so then a partial ordering is to a total ordering as a partial function is to a function}}
{{Requires references|grade=D|msg=One of the reasons this page lacks info is because I can't find a source for any of it!}}
==Definition==
Suppose {{M|f:A\rightarrow B}} is a ''partial [[function]]'', considering {{M|f}} as a [[relation]] this means that, for some {{M|a\in A}}, we have either:
# '''{{M|f}} maps {{M|a}}: '''<ref group="Note">This is my own term. With total orderings any two elements of underlying set of the relation must be comparable. With a total function, {{M|g}}, {{M|g}} must ''map'' every element of its domain to a value. A partial function, doesn't map everything, just as a partial order isn't always comparable</ref> {{M|f(a)}} is "defined" and there exists a {{M|b\in B}} such that {{M|(a,b)\in f}}, which we usually write: {{M|1=f(a)=b}} ({{M|f}} relates {{M|a}} to only {{M|b}}) or
# '''{{M|f}} doesn't map {{M|a}}: ''' {{M|f(a)}} is "undefined" and there '''does not''' exist any {{M|b\in B}} such that {{M|(a,b)\in f}}
==Formulation==
Suppose that {{M|f:A\rightarrow B}} is a partial function, define {{M|\bar{A} }} as follows:
* {{M|1=\bar{A}:=f^{-1}(B):=\{a\in A\ \vert\ \exists b\in B[f(a)=b]\} }} (here {{M|f^{-1}(B)}} denotes the [[pre-image]] of {{M|B}}, which is the set containing all {{M|a\in A}} such that {{M|f}} relates {{M|a}} to a {{M|b\in B}})
Now we get an "[[induced map]]":
* {{M|\bar{f}:\bar{A}\rightarrow B}} that is a (total) [[function]], defined by: {{M|\bar{f}:\bar{a}\mapsto f(\bar{a})}} and we know {{M|f(\bar{a})}} is defined as {{M|\bar{A} }} only contains the elements of {{M|A}} for which {{M|f}} is defined.
We of course get a canonical [[inclusion map]], {{M|i_{\bar{A} }:\bar{A}\rightarrow A}} given by {{M|i_{\bar{A} }:\bar{a}\mapsto\bar{a} }}. Thus we can formulate a partial function like this:
: <m>\xymatrix{ A \ar@{~>}[rr]^f & & B \\ \bar{A} \ar[urr]_{\bar{f} } \ar@{^{(}->}[u]^{i_{\bar{A} } } }</m> where the wiggly line is the partial function.
:* {{Note|This comes from [[Books:An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition]], along with most of the page}}

==Notes==
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==References==
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{{Definition|Set Theory|Elementary Set Theory}}