identity element
Let G be a groupoid, that is a set with a binary operation G×G→G, written muliplicatively so that (x,y)↦xy.
An identity element for G is an element e such that ge=eg=g for all g∈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 |