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?

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.
