Suppose that (C,*,e) is a cyclic monoid generated by the element c of C. We make the following case distinction.
There are k < l with
There are no such k and l.
Let k and l be the smallest
pair (in lexicographical order) with this property.
Then C = {ci | i = 0, 1, ..., l - 1}.
Indeed, for all t 
0, we have cl + t
= ck + t.
ci * cj = ci + j - k - mn
* ck + mn = ci + j - mn.
0, we have cl + t
= ck + t.
Put n = l - k. We shall establish that the map Ck,n -> C sending ci to ci, is an isomorphism. Clearly, it is a bijection.
In C, we have ck + mn =
ck for all m. So, for all i,
j 
N with k + mn 
i + j k + (m + 1)n, we have
This proves that the powers of c in C satisfy the multiplication laws of
Ck,n. This proves that the bijection
Ck,n -> C is a morphism of monoids,
and hence that it is an isomorphism.
The map N -> C given by n -> cn
is readily seen to be an isomorphism from
(N,+,0) to the monoid C.