I understand that the Cartesian product operation is not associative if it is understood as a binary operation.
I.e. (A $\times$ B) $\times$ C != A $\times$ (B $\times$ C).
However when mathematicians write, e.g. A$\times$B$\times$C they actually mean a Cartesian product of arity 3 which is a different operation than the successive application of two binary Cartesian product operators. Since its a ternary or, in general, n-ary operator, associativity does not come into play. Using postfix notation would make that clear. However, since infix is traditionally used in Math, my questions are:
- What kind of notation can be used to signify that the "$\times$" in "A$\times$B$\times$C" actually denotes a ternary operator and not a binary one? Essentially, I am looking for a notation to denote that we are using a single ternary operator "..$\times$..$\times$.." as opposed to two applications of the binary "..$\times$.." operator. I am sure connecting the two "$\times$"s with a curved underline would make that plain but I guess it's hard to do in typography.
- What are some other similar examples of n-ary operators that yield different results than the successive application of their binary counterparts?