Notation for partial function set.

Question

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 $T$?

Answer 1

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.

Answer 2

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}}}

Answer 3

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.