Let $M$ be a magma and $e \in M$.
$e$ is left-neutral if for all $a\in M$,
\[ea = a.\]$e$ is right-neutral if for all $a\in M$,
\[ae = a.\]$e$ is neutral if it is both left-neutral and right-neutral.
$M$ is (left-/right-)unital if it has a (left-/right-)neutral element.
A unital magma has exactly one neutral element.
Proof.βLet $M$ be a unital magma. If both $e$ and $eβ$ are neutral elements of $M$,
\[e = ee' = e'.\]β