Finite product - Knowlpedia

For elements $a_1,\ldots,a_n$ of a monoid with identity $1$ , their finite product is defined recursively by

\prod_{j=1}^{0}a_j=1,\qquad \prod_{j=1}^{n+1}a_j=\left(\prod_{j=1}^{n}a_j\right)a_{n+1}.

Associativity permits changing parentheses. Reordering requires commutativity or a separate argument that the relevant elements commute. The empty-product convention is an identity for multiplication, not a statement that zero factors give the number zero.

Examples

Products of real or complex numbers can be reordered. Products of square matrices generally cannot: $AB$ need not equal $BA$ . The factorial $n!$ is a product of the integers from one to $n$ , so $0!=1$ .