# 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