# What is the name of this set?

What is the standard name for the set of all n-ary functions, where n is a natural number,of some set S, say the reals or the complexes? We have the notation S^S, but that is only the set of 1-ary functions on S. Is there also a name in the literature for the set of all n-ary functions on S, where n is now a positive integer?

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The set of all $n$-ary functions on $S$ can also be called the set of all finitary operations on $S$. –  goblin Jan 2 '14 at 3:45

I startpaged this technical report by Mike Behrisch which uses $O_A$ for the set of all finitary operations and $O_A^{(n)}$ for the set of all $n$-ary operations on a set $A$:

Any mapping $f\in A^{A^n}(n\in\mathbb N)$ is called an $n$-$ary\ operation$ on $A$. The set of
all $finitary\ operations$ on $A$ is $O_A:=\bigcup_{k\in\mathbb N}A^{A^k}$. For a set of operations $F\subseteq O_A$
we denote its $n$-$ary\ part$ by $F^{(n)}:=F\cap A^{A^n}$.

If you need more evidence of the prevalence of this notation, you might check the works cited in Behrisch's report; I haven't done that.

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A function taking $n$ values from $S$ and returning a value in $T$ would be a function from $S \times S \times \cdots \times S = S^n$ to $T$. A standard notation for the set of all functions from $A$ to $B$ is $B^A$. The notation for the set of all functions from $S^n$ to $T$ would then be $T^{S^n}$.

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No, I mean the set of all constants in S, all unary functions in S, all binary functions in S, all 3-ary, ... all in the same set. What is the standard name for that set? –  user107952 Jan 2 '14 at 3:09
$\bigcup_{n \in \mathbb{N}} S^{S^n}$ ? –  Svinepels Jan 2 '14 at 3:10
Yes like that, but a more compact notation. –  user107952 Jan 2 '14 at 3:11
I'm not aware of a standard, compact notation for such a set. –  Svinepels Jan 2 '14 at 3:12
@user107952 Call it $O$ and be done with it. =) –  Pedro Tamaroff Jan 2 '14 at 3:13