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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One of the reasons this page lacks info is because I can't find a source for any of it!

Definition

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

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

Formulation

Suppose that $f:A\rightarrow B$ is a partial function, define $\bar{A}$ as follows:

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

Now we get an "induced map":

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

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

$\xymatrix{ A \ar@{~>}[rr]^f & & B \\ \bar{A} \ar[urr]_{\bar{f} } \ar@{^{(}->}[u]^{i_{\bar{A} } } }$ 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