Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
up vote 2 down vote accepted

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.

share|cite|improve this answer

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 $T^{S^n}$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.