1
$\begingroup$

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?
$\endgroup$
  • 2
    $\begingroup$ 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
2
$\begingroup$

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 $$ \prod_{i=1}^3 A_i. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.