can you simplify values in a monoid? Assume I have a monoid $M$  and I am not guaranteed that all elements have an inverse.

Say I have the property that:

$a^m = a^{m+n}$

Can I claim than it must be that $i=a^n$?

Why or why not?
In general you cannot simplify expressions in monoids this way. Consider a monoid $M = \{i,x\}$, where $i$ is the neutral element and $x$ satisfies $x^2=x$. Then, $x^m=x^{n+m}$ for $n,m \geq 1$, but $i \neq x^n$.

However, in some monoids you can do this (for example, in groups and submonoids of groups).
The answer is yes, if the monoid is a cancellative monoid.  

Here's an example of a non-cancellative monoid under multiplication: $$\{[1],[4]\}\subseteq \mathbb{Z}/6\mathbb{Z}$$

$[4][4]=[4]$, but $[4]\neq [1]$