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 $S$ (i.e., a magma $(S,*)$), 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
• middle neutral or a middle identity if $e * e = a$ for any $a$ in $S$

A neutral element is also termed an identity element.

Facts

• Equality of left and right neutral element: This states that any left neutral element (if it exists) must equal any right neutral element (if it exists). The idea is to consider the product of these elements and show that it equals both of them.
• Binary operation on magma determines neutral element: There can exist at most one two-sided neutral element. Thus, if a neutral element exists, it is unique. This is a corollary of the equality of left and right neutral element.
• 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

Generalizations

• Neutral element for a multiary operation

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