Neutral element

This article defines a property of elements or tuples of elements with respect to a binary operation

Definition

Definition with symbols

Given a binary operation $*$ on a set $S$, an element $e$ in $S$ is termed:

• left neutral or a left identity if $e*a=a$ for any $a$ in $S$
• right neutral or a right identity if $a*e=a$ for any $a$ in $S$
• neutral if it is both left and right neutral

A neutral element is also termed an identity element.

Facts

Any left neutral and right neutral element are equal

The proof of this fact goes as follows: let $e_1$ be a left neutral element and $e_2$ be a right neutral element. Then, the product $e_1*e_2$ is equal to $e_1$ (because $e_2$ is right neutral) and is also equal to $e_2$ (because $e_1$ is left neutral). Hence, $e_1=e_2$.

Some easy corollaries

• If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element
• If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element
• There can exist at most one neutral element. Thus, if a neutral element exists, it is unique

Relation with other properties

Weaker properties

• Idempotent element
• Cancellative element: Any left neutral element is left cancellative, and any right neutral element is right cancellative. Hence, any neutral element is cancellative