On semigroups whose idempotent-generated subsemigroup is aperiodic

by Manuel Delgado,
Vítor H. Fernandes, Stuart Margolis and Benjamin Steinberg

For a pseudovariety ${\bf V}$ of semigroups, we denote by $\ensuremath{{\bf EV}}$ the pseudovariety consisting of all semigroups whose idempotent-generated subsemigroup belongs to ${\bf V}$. For a pseudovariety ${\bf H}$ of groups, we denote by $\mathcal{W}\ensuremath{{\bf H}}$ the smallest pseudovariety of groups containing ${\bf H}$ which is closed under extension and by $\overline{\mathcal{W}\ensuremath{{\bf H}}}$ the pseudovariety of semigroups whose subgroups belong to $\mathcal{W}\ensuremath{{\bf H}}$. The aim of this talk is to prove that for a subpseudovariety ${\bf V}$ of ${\bf A}$, the pseudovariety of all finite aperiodic semigroups, and a non-trivial pseudovariety ${\bf H}$ of groups,

$$\ensuremath{{\bf EV}}\cap \overline{\mathcal{W}\ensuremath{{\bf H... ...$\bigcirc$\rlap{\kern-10pt\raise0,75pt\hbox{\petite m}}}}\ensuremath{{\bf H}}.$$

The technique used in our proof is to combine a result of Graham [2] concerning 0-simple semigroups with iterations of the ${\bf H}$-kernel operator. When the iteration of the ${\bf H}$-kernel operator eventually arrives to the semigroup generated by the idempotents, we call it ${\bf H}$- solvable [1]. Our result shows, in particular, that ${\bf EA}$ is an infinite union of pseudovarieties of the form $(((\ensuremath{{\bf A}}\mathbin{\hbox{$\bigcirc$\rlap{\kern-10pt\raise0,75pt\h... ...irc$\rlap{\kern-10pt\raise0,75pt\hbox{\petite m}}}}\ensuremath{{\bf G}})\cdots$, where $G$ is the pseudovariety of all finite groups. It can not be a finite union, as follows from a construction of Rhodes and Tilson [3] showing that there are semigroups in $(((\ensuremath{{\bf A}}\mathbin{\hbox{$\bigcirc$\rlap{\kern-10pt\raise0,75pt\h... ...irc$\rlap{\kern-10pt\raise0,75pt\hbox{\petite m}}}}\ensuremath{{\bf G}})\cdots$ with $n$ $\ensuremath{{\bf G}}$'s that do not belong to an analogous with $n-1$ ${\bf G}$'s.