Suppose you have a monoid $(M,p,1)$ (viewing it as a triple of a set $M$, operation $p$, and unit $1$). Then for some $m\in M$ we can define a new product $p_m$ in $M$ by $p_m(a,b)=amb$. It's easy to see this is a semigroup.
However, what condition on $m$ will we have a unit relative to $p_m$? If a unit $e$ were to exist, then I suppose $p_m(a,e)=p_m(e,a)=a$, that is, $ame=ema=a$.
What condition on $m$ am I supposed to be getting at? At first I thought we would require that $m$ commute with all of $M$, but the last equality above is giving me a problem. Thanks.