ON THE LOCALITY OF THE PSEUDOVARIETY DG

Abstract

The pseudovariety $\mathbf{DG}$ of all finite monoids all of whose regular $\mathcal{D}$-classes are subgroups is shown to be local, that is, it is verified that the pseudovariety $g\mathbf{DG}$ of finite categories generated by $\mathbf{DG}$ coincides with the pseudovariety $\ell\mathbf{DG}$ of all finite categories whose local monoids all belong to $\mathbf{DG}$. Yet more general statements of this kind are deduced, yielding results such as that, for every prime number $p$, the pseudovariety $\mathbf{DG}_p$ of all finite monoids all of whose regular $\mathcal{D}$-classes are $p$-groups is local, or that the pseudovarieties $\mathbf{DG}_{\mathrm{sol}}$ and $\mathbf{DG}_{\mathrm{nil}}$ of all finite monoids all of whose regular \mbox{$\mathcal{D}$-classes} are, respectively, solvable groups and nilpotent groups are local.