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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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  • $\begingroup$ Obvious remark: for monoids this is trivial (taking $z=1$). $\endgroup$ Commented Jul 10 at 15:07
  • $\begingroup$ Other obvious remark: if one writes $R_z(x)=xz$, this is the requirement of injectivity of $z\mapsto R_z$. And obvious example where it fails: set with an element $0$ and not reduced to $\{0\}$, with constant law $xy=0$ $\forall x,y$. $\endgroup$
    – YCor
    Commented Jul 10 at 15:09
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    $\begingroup$ This seems to be called a right reductive semigroup, at least on Wikipedia and nLab. $\endgroup$ Commented Jul 10 at 15:21
  • $\begingroup$ @NaïmFavier Thanks, that is exactly what i was looking for. $\endgroup$ Commented Jul 10 at 16:38

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A semigroup with this property seems to be called a right reductive semigroup (Wikipedia, nLab).

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