left identity and right identity

Let G be a groupoidPlanetmathPlanetmathPlanetmathPlanetmath. An element eG is called a left identityPlanetmathPlanetmath element if ex=x for all xG. Similarly, e is a right identity element if xe=x for all xG.

An element which is both a left and a right identity is an identity elementMathworldPlanetmath.

A groupoid may have more than one left identify element: in fact the operation defined by xy=y for all x,yG defines a groupoid (in fact, a semigroup) on any set G, and every element is a left identity.

But as soon as a groupoid has both a left and a right identity, they are necessarily unique and equal. For if e is a left identity and f is a right identity, then f=ef=e.

Title left identity and right identity
Canonical name LeftIdentityAndRightIdentity
Date of creation 2013-03-22 13:02:05
Last modified on 2013-03-22 13:02:05
Owner mclase (549)
Last modified by mclase (549)
Numerical id 5
Author mclase (549)
Entry type Definition
Classification msc 20N02
Classification msc 20M99
Related topic IdentityElement
Related topic Unity
Defines left identity
Defines right identity