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?
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.
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.