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Many books classify the standard four arithmetical functions of addition, subtraction, multiplication, and division as binary (in terms of arity). But, "sigma" and "product" notation often writes just one symbol at the front, and indexes those symbols which seemingly makes expressions like $+(2, 3, 4)=9$ meaningful. Of course, we can't do something similar for division and subtraction, since they don't associate, but does the $+$ symbol in the above expression qualify as the same type of expression as when someone writes $2+4=6$? Do addition and multiplication qualify as functions which don't necessarily have a fixed arity, or do they actually have a fixed arity, and thus instances of sigma and product notation should get taken as abbreviation of expressions involving binary functions? Or is the above question merely a matter of perspective? Do we get into any logical difficulties if we regard addition and multiplication as $n$-ary functions, or can we only avoid such difficulties if we regard addition and multiplication as binary?

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Given any binary operation $(x,y) \mapsto F(x,y)$ on a set $X$ you can construct operations with "higher arity", in general in many different ways, e.g.

$g_{3,1}(x,y,z) = F(F(x,y),z)$,
$g_{3,2}(x,y,z) = F(x,F(y,z))$,
$g_{3,3}(x,y,z) = F(y,F(x,z))$,
$g_{3,4}(x,y,z) = F(x,F(y,y))$,

and so forth. If the original binary operation is commutative and/or associative, then some of these induced operations will coincide. In general, tracking down compatibilities between these sort of induced, higher arity operations is something that shows up a lot in category theory (and, especially, higher category theory).

So in particular you can absolutely define three-fold (and higher) subtraction operations; you'll just have more than one of them. (Note also that in most number systems division is not a binary operation, since $a \div 0$ is not defined.)

That's the mathematics of it. The rest of your question seems to be asking about "rules". That depends on further context, which you have not provided. For much of "mainstream" mathematics, the arity of an operation is not of explicit concern or interest. If you are studying universal algebra, or model theory, or other branches of logic, or parts of computer science, it might matter, but there is no uniform set of rules.

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There are no logical difficulties passing back and forth between binary associative operations and their higher-arity extensions. However, a theorem of Sierpinski (Fund. Math., 33 (1945) 169-73) shows that higher-order operations are not needed: every finitary operation may be expressed as a composition of binary operations. The proof is especially simple for operations on a finite set $\rm\:A\:.\:$ Namely, if $\rm\:|A| = n\:$ then we may encode $\rm\:A\:$ by $\rm\:\mathbb Z/n\:,\:$ the ring of integers $\rm\:mod\ n\:,\:$ allowing us to employ Lagrange interpolation to represent any finitary operation as a finite composition of the binary operations $\rm\: +,\ *\:,\:$ and $\rm\: \delta(a,b) = 1\ if\ a=b\ else\ 0\:,\:$ namely

$$\rm f(x_1,\ldots,x_n)\ = \sum_{(a_1,\ldots,a_n)\ \in\ A^n}\ f(a_1,\ldots,a_n)\ \prod_{i\ =\ 1}^n\ \delta(x_i,a_i) $$

When $\rm\:|A|\:$ is infinite one may instead proceed by employing pairing functions $\rm\:A^2\to A\:.$

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Is the theorem every finitary operation with arity of at least 2, or does it also include unary and nullary operations? – Doug Spoonwood Oct 19 '11 at 2:35
@Doug: For any unary operator $f(x)$, we may define a binary operator $g(x,y) := f(x)$. I'll leave extending this to nullary operators as an exercise. – Ilmari Karonen Apr 10 '12 at 14:21
Thanks for not ruining the fun for us, @Ilmari! – The Chaz 2.0 Apr 10 '12 at 15:41

I think it is usually understood as mere notation shorthand, see e.g. wikipedia article giving definition of the sigma notation in terms of expanded series. You can't get in logical difficulties if you define n-ary addition that way, because you can, at your will, either think of it as an operation in itself or a shorthand, as definition is a shorthand for what it defines.

It is not clear however from which point of view are you asking that question. Which "many books"?

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For definiteness, let's think first-order theory of groups. We can change the language by introducing infinitely many function symbols, one of each arity $\ge 2$.

We will then need axioms in order to move freely between products of various arities. These axioms are quite simple, just the usual inductive definition of $n+1$-product in terms of $n$-product and $2$-product. However, we do need infinitely many such axioms.

Occam's Razor suggests that we leave well enough alone, and use arity $3$ multiplication, for example, as an informal abbreviation.

The technical difficulties become much greater if we try to produce a two-sorted theory to handle notions such as $a^n$.

I do not see any demonstrable gain to compensate for the pain of introducing operations of infinitely many arities, or of variable arities. If we really do want what you suggest, we might as well go all the way, and do group theory within a formal set theory. Then all of the usual abbreviations can be given a formal meaning. That is in fact close to the usual mathematical practice, except that the underlying set theory is informal.

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I don't see a necessary connection here, why have you brought up gropus? – Doug Spoonwood Jul 17 '11 at 18:55
@Doug Spoonwood: Your post seemed to be asking about giving an associative (binary) operation multiple arities. Groups is one setting in which we do, at the informal but not the formal level. It seemed reasonable to focus attention on a single class of examples. – André Nicolas Jul 17 '11 at 19:28
I don't see why you talked about groups instead of semigroups. I guess the word "definiteness" threw me off. – Doug Spoonwood Jul 19 '11 at 0:01
@Doug Spoonwood: I actually thought a second or two about the choice. Wanted a class of structures with a familiar axiomatization. Semigroups, though from the axiomatization point of view simpler, and therefore in some sense logically prior, are in fact much less familiar. Many will graduate with a degree in mathematics without having heard the term "semigroup." – André Nicolas Jul 19 '11 at 0:14
Please note that I wrote "without having heard the term." It is possible to work within a structure without giving a name to some wider family that the structure belongs to. – André Nicolas Jul 19 '11 at 0:41
up vote -2 down vote accepted

Addition and multiplication have algebraic properties like commutativity, associativity, x x - +=0, etc. Some of these properties only apply to binary operations, not operations of variable arity. Regarding them as having variable arity would thus lead to a logical problem. Therefore, addition and multiplication have a fixed arity of 2.

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Actually, there are ways to generalize both commutativity and associativity to $n$-ary operators. For commutativity, a reasonable generalization is to require that $\operatorname P(x_1,\dotsc,x_n)=\operatorname P(x_{\sigma_1},\dotsc,x_{\sigma_n})$ for any permutation $\sigma$ of $\{1,\dotsc,n\}$. For associativity, one natural generalization is given by the $n$-ary group axioms. – Ilmari Karonen Apr 10 '12 at 14:38
@IlmariKaronen I do not in any way denying such generalization for commutativity, and associativity. However, do you have a way to generalize all possible algebraic properties? Addition and multiplication satisfy the properties, x 0 +=x, x 1 *=x, x x - +=0, x x / =1, where x / exists, where / and - are particular unary operations. I don't see a generlization happening here, since I don't see how you can have an identity element for a trinary operation, and the cardinality of pairs (y, z) which satisfy x y z +_3 =x, where +_3 indicates 3-ary addition, is infinite. – Doug Spoonwood Apr 10 '12 at 15:26

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