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.