A semigroup can have at most one zero

Show that a semigroup can have at most one zero.


Recall that an element z \in S is called a left zero if zs = z for all s \in S and a right zero if sz = z for all s \in S, and a zero if it is both a left and a right zero.

Suppose z is a left zero and w a right zero. Then w = zw = z. In particular, if z and w are zeros in S, then z = w.

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