Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the Green's R relation. 


Is the pseudovariety $\textbf{BG}\cap\textbf{R}$ well known in the field of semigroup&monoid theory? Is there somewhere in the literature a concrete characterization of this pseudovariety (hopefully) in terms of formal languages recognized by it's syntactic monoids? See [2].


I already know that this pseudovariety is defined by pseudoidentities 
$[(xy)^{\omega}x = (xy)^w$, $(x^{\omega}y^{\omega})^{\omega}=(y^{\omega}x^{\omega})^{\omega}]$.


[1] JE Pin, BG=PG: A Success Story


[2] (mathoverflow.net/users/16758), Finite variation and idempotent languages and automata., [Finite variation and idempotent languages and automata.](https://mathoverflow.net/questions/71935)



The intersection of Block Groups and R-trivial (finite) monoids
I believe the answer is the pseudo variety of J-trivial monoids. Each regular J-class of an R-trivial monoid is a left zero semigroup. The block group condition allows only the trivial left zero semigroup. So each regular J-class is trivial. Thus all J-classes are trivial. The corresponding languages are the piecewise testable ones.


