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 .