Prove that every infinite cyclic semigroup is isomorphic to the semigroup of positive natural numbers under addition.
First, let’s clarify some terminology. If is a semigroup and , then there is a semigroup homomorphism given by . The image of is called the cyclic subsemigroup of generated by , and if is surjective, then we say is cyclic with as a generator. If has an (unique) identity element , then the homomorphism extends to a mapping by and this is also a semigroup homomorphism. The image of is called the cyclic submonoid of generated by .
First, it is clear that is an infinite cyclic semigroup.
Suppose now that is an infinite cyclic semigroup with a generator. Now given by is a surjective semigroup homomorphism. We only need to show that is injective. To that end, suppose we have for some , so that . Suppose, without loss of generality, that ; say , so that . But now for all , by the division algorithm we have for some and . Then . Indeed, for all positive natural numbers , we have . In particular, , hence , is finite- a contradiction. Thus , and so is injective. So .