# Definition:Operation/Binary Operation/Product

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## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be the operation on $\struct {S, \circ}$.

Let $z = x \circ y$.

Then $z$ is called the **product** of $x$ and $y$.

This is an extension of the normal definition of product that is encountered in conventional arithmetic.

### Left-Hand Product

Let $x$ and $y$ be elements which are operated on by a given operation $\circ$.

The **left-hand product of $x$ by $y$** is the product $y \circ x$.

### Right-Hand Product

Let $x$ and $y$ be elements which are operated on by a given operation $\circ$.

The **right-hand product of $x$ by $y$** is the product $x \circ y$.

## Also known as

The **product** of $a$ and $b$ is sometimes seen referred to as their **sum**.

This can be confusing and is therefore endorsed on this site only when referring to addition.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$. Definition and examples of semigroups - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups