This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success.
Let $M$ be a monoid and $u\in M$. Suppose there exists a unique $z\in M$ such that $uzu=u$. Does it imply that $u$ is an invertible element of $M$, with $u^{-1}=z?$
The only thing I see is that it implies that $z=zuz,$ since $u(zuz)u=(uzu)zu=uzu=u.$
So $z$ must be a (unique) von Neumann generalized inverse of $u$. But this is far from enough...