\begin{definition}[Definition:Product Notation (Algebra)/Propositional Function]
Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.
Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
Let $\map R j$ be a propositional function of $j$.
Then we can write:
:$\ds \prod_{\map R j} a_j = \text{ The product of all $a_j$ such that $\map R j$ holds}$.
If more than one propositional function is written under the product sign, they must ''all'' hold.
Such an operation on an ordered tuple is known as a '''product'''.
Note that the definition by inequality form $1 \le j \le n$ is a special case of such a propositional function.
Also note that the definition by index form $\ds \prod_{j \mathop = 1}^n$ is merely another way of writing $\ds \prod_{1 \mathop \le j \mathop \le n}$.
Hence ''all'' instances of a '''summation''' can be expressed in terms of a propositional function.
\end{definition}