MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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] (, Finite variation and idempotent languages and automata., Finite variation and idempotent languages and automata.

share|cite|improve this question
up vote 2 down vote accepted

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.

share|cite|improve this answer
BTW, your pseudoidentiy also says J-trivial. The second one says each J-class has a unique idempotent and the first forces aperiodic. – Benjamin Steinberg Aug 9 '11 at 15:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.