I am currently studying construction of the Grothendieck group of a commutative monoid $M$. I was looking for an example of a monoid that is torsion, namely, I have the following query.
Does there exist a monoid $M$ (written multiicatively with identity element denoted by $1$) that is not a group such that $x \in M$ implies $x^n = 1$ for some $n \in \mathbb{N}$?