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I have already asked a similar question, but it appears I must ask a more fundamental question about Cartesian products.

In my experience, we normally notate in accordance with this expression:

$$(s_1,s_2,\ \dots , s_n) \in S^n \tag1$$

However, some mathematicians mean something else by the Cartesian product:

$$((s_1,s_2),s_3) \in S^3$$

Or more generally (and less aesthetically pleasing):

$$((\cdots(s_1,s_2),s_3),s_4),\ \dots \ ), s_n) \in S^n \tag2$$

Now, perhaps everyone means this by a Cartesian product, but typically, context requires (and clearly implies) the use of $(1)$. Given the existence of a bijection between them, using $(1)$ in practice despite defining Cartesian products as $(2)$ is fine.

However, this raises problems. Firstly, how does one clearly specify which one is using? There are probably multiple ways and I am not asking for your opinions on which is best; simply, I want to know what options I have. In many cases, it might not be practically needed, but in some cases, it is. Especially when the use of $(1)$ and $(2)$ co-occur, as in the case that motivated my linked-to question. A set with elements that are tuples of tuples (of tuples, etc.) means that you want some elements to co-exist in the same, immediate tuple, and some to not; thus using both $(1)$ and $(2)$ simultaneously.

How could one go about unambiguously notating such a thing? Or more generally, how does one specify exactly how one is using the Cartesian product?

user110391
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  • If you define $X^n$ as $n$-tuples then $X^2\times X$ and $X^3$ and $X\times X^2$ are different objects you can distinguish. It's just that in most cases in practice we use them interchangeably through the obvious isomorphism, which is used implicitly and not mentioned. – Michal Adamaszek Jun 13 '23 at 05:48
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    Your question is really about specifying tuples. – copper.hat Jun 13 '23 at 05:48
  • I'm not aware of a good notation, however, I think of $A^B$ as the maps from $B$ to $A$ so $S^n$ would signal to me maps from $n$ (which in set theory equals ${0,\dots,n-1}$) to $S$ hence tuples of the form $(s_1,\dots,s_n)$. The other thing I'd probably notate like $(S\times S)\times S$. – fweth Jun 13 '23 at 07:34

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