Question

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].

All I already know is 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., http://mathoverflow.net/questions/71935

Answer 1

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.