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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 '14 at 20:15
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    $\begingroup$ In some sense, $(T + 1)^S$. $\endgroup$ – Zhen Lin Apr 6 '14 at 20:19
  • $\begingroup$ @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.stm Apr 6 '14 at 20:20
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    $\begingroup$ @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 Lin Apr 6 '14 at 20:35
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    $\begingroup$ Take a partial function and turn it into a total function by sending elements outside the domain to the new element. $\endgroup$ – Zhen Lin Apr 7 '14 at 7:54
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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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Z-notation uses this: ⇸ (\char"21F8 in latex)

See, e.g., http://staff.washington.edu/jon/z-lectures/structure.html

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