Condition for Group given Semigroup with Idempotent Element

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let there exist an idempotent element $e$ of $S$ such that for all $a \in S$:
 * there exists at least one element $x$ of $S$ satisfying $x \circ a = e$
 * there exists at most one element $y$ of $S$ satisfying $a \circ y = e$.

Then $\struct {S, \circ}$ is a group.

Proof
Let $a$ be arbitrary.

We have:

So $e$ is a right identity.

Again, let $a$ be arbitrary.

Let $x \in S$ be such that $x \circ a = e$.

We have:

So $e$ is a left identity.

We note the meaning of the criterion:
 * there exists at least one element $x$ of $S$ satisfying $x \circ a = e$

As we now know that $e$ is a left identity, the above means that $x$ is a left inverse for $a$ in $S$.

To summarise, we have an algebraic structure $\struct {S, \circ}$:
 * $(1): \quad$ which is closed, from
 * $(2): \quad$ which is associative, from
 * $(3): \quad$ which has a left identity
 * $(4): \quad$ for which every element has a left inverse.

That is, $\struct {S, \circ}$ fulfils all the left group axioms.

Hence the result.