# Definition:Product Notation (Algebra)/Index

< Definition:Product Notation (Algebra)(Redirected from Definition:Indexed Product)

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## Definition

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 $S$.

The composite is called the **product** of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

- $\displaystyle \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$

## Multiplicand

The set of elements $\left\{{a_j \in S: 1 \le j \le n, R \left({j}\right)}\right\}$ is called the **multiplicand**.

## Notation

The sign $\displaystyle \prod$ is called **the product sign** and is derived from the capital Greek letter $\Pi$, which is $\mathrm P$, the first letter of **product**.

## Also see

- Results about
**products**can be found here.

## Historical Note

The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 18$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers