There is a standard notation for the set of all functions between $S$ and $T$, namely $T^S$. Is there a similar notation for the set of all partial functions between $S$ and $T$?
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$\begingroup$ Not that I know of. $\endgroup$– Asaf Karagila ♦Apr 6, 2014 at 20:15
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5$\begingroup$ In some sense, $(T + 1)^S$. $\endgroup$– Zhen LinApr 6, 2014 at 20:19
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$\begingroup$ @AsafKaragila Wow. By itself, this is a very useless comment, but it probably carries some weight because you know a lot of mathematics and are into logic and set theory. $\endgroup$– k.stmApr 6, 2014 at 20:20
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3$\begingroup$ @Student There is a natural bijection between $(T + 1)^S$ and the set of partial functions $S \rightharpoonup T$. Or you can just count. $\endgroup$– Zhen LinApr 6, 2014 at 20:35
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2$\begingroup$ Take a partial function and turn it into a total function by sending elements outside the domain to the new element. $\endgroup$– Zhen LinApr 7, 2014 at 7:54
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3 Answers
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I wouldn't call this "standard", but I like this notation for the set of all partial maps from $X$ to $Y$, from Davey and Priestley's Introduction to Lattices and Order:
$$(X \multimap \mspace{-2mu} \to Y)$$
In LaTeX:
(X \multimap \mspace{-2mu} \to Y)
Update (2018-09-25): I've decided that I don't like the above notation after all. May I suggest $Y^{\subseteq X}$ for the set of all partial functions? This nicely complements $Y^X$ and $Y^{(X)}$ (the set of all finitely supported functions). Also, from Duistermaat and Kolk's Multidimensional Real Analysis I: $f : X \supset\!\to Y$ means $f$ is a partial function from $X$ to $Y$ (actually, the authors use this to denote a total function from some subset of $X$ into $Y$, when they don't want to explicitly specify the domain). I like this because the "$\supset$" part of the symbol suggests that $X$ is a superset of something.
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I suggest the notation
$$ A \rightharpoonup B = \bigcup_{C \subseteq A} (C \rightarrow B) $$
where $\rightharpoonup$ is \rightharpoonup in TeX. A Google Scholar search yields many results using this notation for partial functions.
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$\begingroup$ I like this notation: $A \rightharpoonup B$. Can anyoneone point to a reference where this notation is used? $\endgroup$– balageJan 7, 2021 at 12:39
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1$\begingroup$ @balage See the link I added to the answer. $\endgroup$ Jan 7, 2021 at 19:24
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Z-notation uses this: ⇸ (\char"21F8 in latex)
See, e.g., http://staff.washington.edu/jon/z-lectures/structure.html
How? Just add this:
% Partial function character from Z-notation
\newcommand\pfun{\mathrel{\ooalign{\hfil$\mapstochar$\hfil\cr$\to$\cr}}}
