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I have two similar questions, one about the H-trivial monoids and one about the R-trivial monoids.

  1. I cannot see the reason why H-trivial monoids, i.e., the monoids where H induced classes are singletons, coincide with the variety A of aperiodic monoids, also characterized as the monoids that satisfy the monoid equation $x^\omega x=x$.
  2. Similarly, I don't understand why R-trivial monoids, i.e., the monoids where R induced classes are singletons, coincide with monoids that satisfy the monoid equation $(xy)^\omega x=(xy)^\omega$.

Here

  1. $x^\omega$ is defined as the limit $\lim\limits_{k\rightarrow\infty} x^{k!}$.
  2. the relation $R$ is defined as $xRy \iff xM=yM$.
  3. the relation $L$ is defined as $xLy \iff Mx=My$.
  4. the relation $H$ is defined as $xHy\iff xRy \land xLy$.
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