# Notation for partial function set.

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 $T$?

• Not that I know of. Apr 6, 2014 at 20:15
• In some sense, $(T + 1)^S$. Apr 6, 2014 at 20:19
• @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. Apr 6, 2014 at 20:20
• @Student There is a natural bijection between $(T + 1)^S$ and the set of partial functions $S \rightharpoonup T$. Or you can just count. Apr 6, 2014 at 20:35
• Take a partial function and turn it into a total function by sending elements outside the domain to the new element. Apr 7, 2014 at 7:54

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.

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.

• I like this notation: $A \rightharpoonup B$. Can anyoneone point to a reference where this notation is used? Jan 7, 2021 at 12:39
% Partial function character from Z-notation
\newcommand\pfun{\mathrel{\ooalign{\hfil$\mapstochar$\hfil\cr$\to$\cr}}}