left identity and right identity

Let $G$ be a groupoid. An element $e\in G$ is called a left identity element if $ex=x$ for all $x\in G$. Similarly, $e$ is a right identity element if $xe=x$ for all $x\in G$.

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 operationdefined by $xy=y$ for all $x,y\in G$ 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 $f=ef=e$.