identity element

Let G be a groupoidPlanetmathPlanetmathPlanetmathPlanetmath, that is a set with a binary operationMathworldPlanetmath G×GG, written muliplicatively so that (x,y)xy.

An identity elementMathworldPlanetmath for G is an element e such that ge=eg=g for all gG.

The symbol e is most commonly used for identity elements. Another common symbol for an identity element is 1, particularly in semigroupPlanetmathPlanetmath 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