Notation for a string of objects

Question

Is there a commonly used notation for a string of objects? The particular situation I am interested in is a string of elements from a Boolean algebra. The elements in the string may be expressions using the Boolean operations $\wedge$ and $\vee$. When this situation occurs I enclose such an element in parentheses. A typical example of such a string is $$(a_{1} \wedge b_{1}) \ldots (a_{m} \wedge b_{1}) (a_{1} \wedge b_{2}) \ldots (a_{m} \wedge b_{2}) \ldots (a_{1} \wedge b_{n}) \ldots (a_{m} \wedge b_{n})$$ What I would like is something like a summation sign so that I could save space and improve clarity. So that this string could be written in a manner similar to $$\sum_{i = 1}^{m}\sum_{j = 1}^{n} (a_{i} \wedge b_{j}) \text{,}$$ but without the implied operation of addition.

Answer 1

A proposal is to use:

$\{ (a_i \land b_j)_{1 \le i \le n, 1 \le j \le m} \}$.

Answer 2

Hint: A commonly used notation for creating strings is concatenation of letters taken from an alphabet $V$.

We could then write \begin{align*} &(a_{1} \wedge b_{1}) \ldots (a_{m} \wedge b_{1}) (a_{1} \wedge b_{2}) \ldots (a_{m} \wedge b_{2}) \ldots (a_{1} \wedge b_{n}) \ldots (a_{m} \wedge b_{n})\\ &\qquad=\prod_{l=1}^n\prod_{k=1}^m(a_{k} \wedge b_{l}) \end{align*} where non-commutativity of concatenation has to be appropriately addressed.