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:


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

Vacuous Product
Take the product:
 * $$\prod \limits_{\Phi \left({j}\right)} a_j$$

where $$\Phi \left({j}\right)$$ is a propositional function of $$j$$.

Suppose that there are no values of $$j$$ for which $$\Phi \left({j}\right)$$ is true.

Then $$\prod \limits_{\Phi \left({j}\right)} a_j$$ is defined as being $$1$$. Beware: not zero.

This summation is called a vacuous product.

This is most frequently seen in the form:
 * $$\prod_{j=m}^n a_j = 1$$

where $$m > n$$.

In this case, $$j$$ can not at the same time be both greater than or equal to $$m$$ and less than or equal to $$n$$.

Compare vacuous truth.

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:


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