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.

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 $y \circ x$.

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 $x \circ y$.

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.

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