Have the semigroups with the following cancellation property been studied?

**Property**: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.

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or
monoidsthis is trivial (taking $z=1$). $\endgroup$right reductive semigroup, at least on Wikipedia and nLab. $\endgroup$