Condition for Group given Semigroup with Idempotent Element

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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




Sources