I do not have enough reputation to comment, so I had to a make a post to ask this question. If this does not site standards feel free to remove it.
What condition on $m$ turns this semigroup into a monoid?
So in the above post user23211 (at the very bottom of the page) mentions that if $(M, p)$ is not a monoid then $(M, p_m)$ is also not a monoid. However, I don't seem to understand the reasoning there. Clearly $em$ is a left identity and $me$ is a right identity.
However, how does it follow that $me = em$ as we no longer have the fact that $(M,p)$ has a unit?