## Every infinite cyclic semigroup is isomorphic to the positive natural numbers under addition

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 .

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