Definition:Product Notation (Algebra)

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:

$$\prod \limits_{j=1}^n a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$$

Alternatively:

$$\prod \limits_{1 \le j \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$$

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

$$\prod \limits_{\Phi \left({j}\right)} a_j =$$ The product of all $$a_j$$ such that $$\Phi \left({j}\right)$$ holds.

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:

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