# Why H-trivial monoids correspond to the variety of aperiodic monoids

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 $$xHy\iff xRy \land xLy$$.