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]: [Note 1] [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.

Notes

  1. This is my own term. With total orderings any two elements of underlying set of the relation must be comparable. With a total function, [ilmath]g[/ilmath], [ilmath]g[/ilmath] 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

References