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.

Title	identity element
Canonical name	IdentityElement
Date of creation	2013-03-22 12:49:07
Last modified on	2013-03-22 12:49:07
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	9
Author	mclase (549)
Entry type	Definition
Classification	msc 20A05
Classification	msc 20N02
Classification	msc 20N05
Classification	msc 20M99
Synonym	neutral element
Related topic	LeftIdentityAndRightIdentity
Related topic	Group