nLab
identity element

Identity elements

Definitions

Given an operation *:X×YY, an element e of X is called a left identity for * if e*a=a for every element a of Y. That is, the map YY given by e* is the identity function on Y.

If *:Y×XY, then there is a similar concept of right identity.

If *:X×XX, then e is a two-sided identity, or simply identity, if it is both a left and right identity.

Historically, identity elements (as above) came first, then identity functions, and then identity morphisms. These are all the same basic idea, however: an identity morphism is an identity element for the operation of composition.

An identity is sometimes called a unit (although that term also has a broader meaning, and an operation that has an identity element is called unital or unitary. In particular, a magma whose underlying operation *:X×XX has an identity is called a unital magma or a unitary magma.

Similarly, a unit law is the statement that a given operation has an identity element. In higher category theory, we generalise from the property of uniticity/unitality to the structure of a unitor.

Revised on April 29, 2013 16:21:51 by Urs Schreiber (89.204.130.252)