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?