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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