## You are here

Homeidentity element

## Primary tabs

# identity element

Let $G$ be a groupoid, that is a set with a binary operation $G\times G\to G$, written muliplicatively so that $(x,y)\mapsto xy$.

An *identity element* for $G$ is an element $e$ such that $ge=eg=g$ for all $g\in G$.

The symbol $e$ is most commonly used for identity elements. Another common symbol for an identity element is $1$, particularly in semigroup theory (and ring theory, considering the multiplicative structure as a semigroup).

Groups, monoids, and loops are classes of groupoids that, by definition, always have an identity element.

Related:

LeftIdentityAndRightIdentity, Group

Synonym:

neutral element

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

20A05*no label found*20N02

*no label found*20N05

*no label found*20M99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections