left identity and right identity
Let be a groupoid. An element is called a left identity element if for all . Similarly, is a right identity element if for all .
An element which is both a left and a right identity is an identity element.
A groupoid may have more than one left identify element: in fact the operation defined by for all defines a groupoid (in fact, a semigroup) on any set , 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 is a left identity and is a right identity, then .
Title | left identity and right identity |
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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 |