Definition 1

Under a law of composition $\top$ on a set $E$ an element $e$ of $E$ is called an identity element if, for all $x \in E$, $e \top x = x \top e = x$.

There exists at most one identity element under a given law $\top$, for if $e$ and $e'$ are identity elements then $e = e \top e' = e'$. An identity element is permutable with every element: it is a central element.