Partial-function

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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

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Definition

Suppose [ilmath]f:A\rightarrow B[/ilmath] is a partial function, considering [ilmath]f[/ilmath] as a relation this means that, for some [ilmath]a\in A[/ilmath], we have either:

1. [ilmath]f[/ilmath] maps [ilmath]a[/ilmath]: [ilmath]f(a)[/ilmath] is "defined" and there exists a [ilmath]b\in B[/ilmath] such that [ilmath](a,b)\in f[/ilmath], which we usually write: [ilmath]f(a)=b[/ilmath] ([ilmath]f[/ilmath] relates [ilmath]a[/ilmath] to only [ilmath]b[/ilmath]) or
2. [ilmath]f[/ilmath] doesn't map [ilmath]a[/ilmath]: [ilmath]f(a)[/ilmath] is "undefined" and there does not exist any [ilmath]b\in B[/ilmath] such that [ilmath](a,b)\in f[/ilmath]

Formulation

Suppose that [ilmath]f:A\rightarrow B[/ilmath] is a partial function, define [ilmath]\bar{A} [/ilmath] as follows:

• [ilmath]\bar{A}:=f^{-1}(B):=\{a\in A\ \vert\ \exists b\in B[f(a)=b]\}[/ilmath] (here [ilmath]f^{-1}(B)[/ilmath] denotes the pre-image of [ilmath]B[/ilmath], which is the set containing all [ilmath]a\in A[/ilmath] such that [ilmath]f[/ilmath] relates [ilmath]a[/ilmath] to a [ilmath]b\in B[/ilmath])

Now we get an "induced map":

• [ilmath]\bar{f}:\bar{A}\rightarrow B[/ilmath] that is a (total) function, defined by: [ilmath]\bar{f}:\bar{a}\mapsto f(\bar{a})[/ilmath] and we know [ilmath]f(\bar{a})[/ilmath] is defined as [ilmath]\bar{A} [/ilmath] only contains the elements of [ilmath]A[/ilmath] for which [ilmath]f[/ilmath] is defined.

We of course get a canonical inclusion map, [ilmath]i_{\bar{A} }:\bar{A}\rightarrow A[/ilmath] given by [ilmath]i_{\bar{A} }:\bar{a}\mapsto\bar{a} [/ilmath]. Thus we can formulate a partial function like this:

[ilmath]\xymatrix{ A \ar@{~>}[rr]^f & & B \\ \bar{A} \ar[urr]_{\bar{f} } \ar@{^{(}->}[u]^{i_{\bar{A} } } }[/ilmath] where the wiggly line is the partial function.

• This comes from Books:An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition, along with most of the page