Binary operation on magma determines neutral element

Statement

Suppose $(S,*)$ is a magma (set $S$ with binary operation $*$). Then, if there exists a neutral element for $*$ (i.e., an element $e$ such that $e * a = a * e = a$ for all $a \in S$), the element $e$ is uniquely determined by $*$.

In other words, a magma can have at most one two-sided neutral element.

Facts used

Equality of left and right neutral element

Related facts

In the case that $*$ is associative, this says that the identity element (neutral element) of a monoid is completely determined by the binary operation. This yields the fact that monoids form a non-full subcategory of semigroups.