Definition:Product Notation (Algebra)

Definition
Let $\left({S, \times}\right)$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.

Let $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

Then the composite is called the product of $\left({a_1, a_2, \ldots, a_n}\right)$, and is written:


 * $\displaystyle \prod_{j \mathop = 1}^n a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

This can also be written:
 * $\displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$

If $\Phi \left({j}\right)$ is a propositional function of $j$, then we can write:


 * $\displaystyle \prod_{\Phi \left({j}\right)} a_j = \text{ The product of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

Note that $1 \le j \le n$ is in fact a special case of such a propositional function, and that $\displaystyle \prod_{j \mathop = 1}^n$ is merely another way of writing $\displaystyle \prod_{1 \mathop \le j \mathop \le n}$.

If an infinite number of values of $j$ satisfy the propositional function $\Phi \left({j}\right)$, then the precise meaning of $\displaystyle \prod_{\Phi \left({j}\right)} a_j$ is:


 * $\displaystyle \prod_{\Phi \left({j}\right)} a_j = \left({\lim_{n \to \infty} \prod_{{\Phi \left({j}\right)} \atop {-n \mathop \le j \mathop < 0}} a_j}\right) + \left({\lim_{n \to \infty} \prod_{{\Phi \left({j}\right)} \atop {0 \mathop \le j \mathop \le n}} a_j}\right)$

provided that both limits exist.

If either limit does fail to exist, then the infinite product does not exist.

Note also that if more than one propositional function is written under the $\displaystyle \prod$ sign, they must all hold.

Cartesian Product of Sets
The following notation is also customary.

Let $\left \langle {S_n} \right \rangle$ be a sequence of sets.

The cartesian product of $\left \langle {S_n} \right \rangle$ can be written as:


 * $\displaystyle \prod_{k \mathop = 1}^n S_k = \left\{{\left({x_1, x_2, \ldots, x_n}\right): \forall k \in \N^*_n: x_k \in S_k}\right\}$

Also see

 * Definition:Summation