Definition:Operation/Binary Operation/Product

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.