## A semigroup can have at most one identity element

Show that a semigroup can have at most one identity element.

Recall that a left identity is an element $e$ such that $es = s$ for all $s \in S$, and a right identity an element $f$ such that $sf = s$ for all $s \in S$. An identity is an element which is both a left and a right identity.

Suppose $e \in S$ is a left identity and $f \in S$ a right identity. Then $e = ef = f$. In particular, if $e$ and $f$ are identities, then $e = f$.

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