1
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ BTW, your pseudoidentiy also says J-trivial. The second one says each J-class has a unique idempotent and the first forces aperiodic. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 9, 2011 at 15:14

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .