When dealing with multiple sets it is common practice to use convenient notations $\displaystyle\bigcup_{k=1}^nA_k$ and $\displaystyle\bigcap_{k=1}^nA_k$ to mean the union and intersection respectively of sets $A_1, A_2,\dots,A_n$.
I haven't ever seen similar notation applied to the membership ("element of") symbol - i.e. something like $\displaystyle a_k\mathop\in_{k=1}^n A_k$ - though it seems that it would be as convenient and as implicitly understandable as the notation above.
For example, this could allow us to define a Cartesian product very succinctly (and still very understandably) as:
$$A_1\times A_2\times\cdots\times A_n = \left\{(a_1, a_2,\dots,a_n):a_k\mathop\in_{k=1}^n A_k \right\}$$
Or to stretch a point:
$$\mathop\times_{k=1}^n A_k=\left\{(a_1, a_2,\dots,a_n):a_k\mathop\in_{k=1}^n A_k \right\}$$
Is there a reason or convention for the fact that this kind of notation is commonly used for some operators while not others? I can see, for example, that $\cup$ and $\cap$, unlike $\in$ but like $\sum$ and $\prod$, represent operations repeatedly performed on objects of the same kind. Does it have to do with this?
\bigcap
instead of\displaystyle\mathop\cap
. $\endgroup$\prod
to denote the product of a sequence, in this case the cartesian product but the notation is used for numbers as well. $\prod\limits_{k=1}^n A_k$. $\endgroup$