I have seen that being used like in n-ary Cartesian product.
But we know that Cartesian product is not associative so $(A \times B) \times C \neq A \times (B \times C)$ and therefore $A \times B \times C$ is not well-defined. (Like the cross product, we need to have the parentheses.)
EDIT:
My main concern is how all of these things can be defined (listed below)? Are all of them definable at all?
As long as we are dealing with only two sets, $A \times B$ can only mean one thing.
But ambiguities arise when the number of sets involved increases:
(For compactness, I will use the $AB$ notation instead of $A\times B$)
3 sets: $$(AB)C; \quad ABC; \quad A(BC)$$
4 sets: $$ABCD$$ $$(AB)CD; \quad A(BC)D; \quad AB(CD)$$ $$(ABC)D; \quad A(BCD)$$ $$((AB)C)D; \quad A((BC)D)$$ $$(A(BC))D; \quad A(B(CD))$$
And so on.
Up to now, It has been made clear that:
- Sets like $X_1 \cdots X_n$ are accepted to have elements of the form $(x_1,\dots,x_n)$ -- Like $\mathbb{R}^n$.
- Sets that have clearly shown the order of operation, have elements of the form similar to the parentheses placement. For example $(x_1,((x_2,x_3),x_4)) \in A((BC)D)$. (Am I right?)
But again some of the items of the above list remain ambiguous. Like $(ABC)D$.
Can we define this?
Can we set some rule for defining even more complicated combinations?
In general we do not have $\langle\langle a,b\rangle,c\rangle=\langle a,\langle b,c\rangle\rangle$ so definitions like $A\times B\times C:=(A\times B)\times C$ or $A\times B\times C:=A\times(B\times C)$ are not canonical.
A nice solution for that is defining $X_1\times\cdots\times X_n$ as the set of functions $f:\{1,\dots,n\}\to\bigcup_{i=1}^nX_i$ that satisfy $f(i)\in X_i$ for every $i\in\{1,\dots,n\}$.
If $f$ is an element of that set and $f(i)=x_i\in X_i$ for $i=1,\dots,n$ then $\langle x_1,\dots,x_n\rangle$ can be looked at as a nice notation for that function, and nothing more than that.
This also works for other index sets:
$$\prod_{i\in I}X_i=\{f\mid f\text{ is a function }I\to\bigcup_{i\in I}X_i\text{ s.t. }\forall i\in I[f(i)\in X_i]\}$$
The projection $p_i:\prod_{i\in I}X_i\to X_i$ is prescribed by $f\mapsto f(i)$.