I have this four part question.
a) Let $M$ be a monoid generated by a single element $a$. Call $m \in \mathbb{Z}^+$ a period of $M$ if $a^n = a^{n+m}$ for some $n \geq 0$. Let $p$ be the smallest period of $M$, if exists. Prove that any other period $m$ of $M$ is a multiple of $p$.
This was easily done by contradiction so I will forego writing it here.
b) If $M = \langle a \rangle$ and $p$ is as in $(a)$, prove that $a^n = a^{n+m}$ implies $a^n=a^{n+p}=...=a^m$ I am really confused about this one. Is that a typo and the last term is supposed to be $a^{n+m}$? If that is the case I proved this as part of (a) and have nothing to worry about. Otherwise, I have no idea how to proceed.
c) Consider the set $\mathbb{Z}_{\geq 0}$ as an additive monoid. Describe all congruences on $\mathbb{Z}_{\geq 0}$
I have a feeling that there is only one congruence, but I'm struggling to prove it.
d) Prove that any monoid $M$ generated by a single element is either isomorphic ot $\mathbb{Z}_{\geq 0}$ or has a presentation of the form $\langle a | a^n = a^{n+p} \rangle$. I have no idea how to do this one. I guess that the idea is if $M$ has a period then it has a least one $p$. If it doesn't then we probably consider the congruences (single congruence?) from (c)
(b) This is indeed a typo.
(c) and (d). Hint: The additive monoid $\mathbb{Z}_{\geq 0}$, better known as $\mathbb{N}$, has infinitely many congruences. Let $f_{n,p}$ be the unique monoid morphism from $\mathbb{N}$ onto $\langle a \mid a^n = a^{n+p} \rangle$ such that $f_{n,p}(1) = a$. The corresponding congruence $\equiv_{n,p}$ is defined by $$ s \equiv_{n,p} t \text{ if and only if } f_{n,p}(s) = f_{n,p}(t). $$ For instance $9 \equiv_{7,3} 12 \equiv_{7,3} 15 \equiv_{7,3} \dotsm$, but $6 \not\equiv_{7,3} 9$.