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:

  1. 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.
  2. What are some other similar examples of n-ary operators that yield different results than the successive application of their binary counterparts?
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    $\begingroup$ Although $(A\times B)\times C\neq A\times (B\times C)$, there is an incredibly obvious bijection between them that preserves every interesting property about them. As such we usually say $(A\times B)\times C\equiv A\times (B\times C)$ because for all intents and purposes they are the same to us. $\endgroup$ – JMoravitz Sep 27 '17 at 1:04

For the first question, you could write $$ \times(A,B,C). $$ A more standard notation would be to consider the sets to be indexed by an ordered set, such as $A_1,A_2,A_3$ are indexed by $\{1,2,3\}$, and then to consider $$ \prod_{i=1}^3 A_i. $$


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