Exhibit a subsemigroup of a cyclic semigroup which is not cyclic.
Consider the semigroup of natural numbers under addition. Certainly is cyclic with generator 1. Define by .
Certainly is nonempty, since . Moreover, if , then . So is a subsemigroup of a cyclic semigroup.
Now suppose is cyclic, with generator . Now , so for some . Certainly . If , then we have , a contradiction. So , and we have . So , and thus . That is, is generated by . But we also have , so that for some , a contradiction.
So is not cyclic.