can you simplify values in a monoid?

Question

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?

Answer 1

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).

Answer 2

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 $[4]\neq [1]$

Answer 3

The answer is no : take in $\mathcal{M}_2(\mathbb{R})$ with the regular matrix multiplication $$ A= \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right)$$ $$ A^{m+n}=A^m \neq I , \qquad \forall m,n>0$$