Do addition and multiplication have arity? 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?
 A: 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 $\rm\:A^2\to A\:.$
A: 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.
A: 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.
A: 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"? 
A: 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.
