Left neutral element is unique idempotent with left inverse in semigroup

Statement

Suppose $(S,*)$ is a semigroup (in other words, $*$ is an associative binary operation $S\times S\to S$) and $e$ is a left neutral element for the magma, i.e., we have:

$e*a=a=\ \forall \ a\in S$

Suppose $u$ is an idempotent element of $S$ (i.e., $u*u=u$) with a left inverse $v$ with respect to $e$, so $v*u=e$. Then, $u=e$.

Related facts

• Left neutral element is unique in semigroup if every element has a left inverse

Proof

Given: Semigroup $(S,*)$. Element $e\in S$ such that $e*a=a\ \forall a\in S$, and for all $a\in S$. Idempotent $u\in S$ and element $v\in S$ such that $v*u=e$.

To prove: $u=e$

Proof: Consider the product $v*u*u$. Parenthesized as $(v*u)*u$, it simplifies to $e*u=u$. Parenthesized to $v*(u*u)$, it simplifies to $v*u=e$. Thus, $u=e$.