Pointlike sets for varieties determined by groups

Abstract

For a variety of finite groups $\mathbf{H}$, let $\overline{\mathbf{H}}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf{H}$. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to $\overline{\mathbf{H}}$. Our characterization is effective whenever $\mathbf{H}$ has a decidable membership problem. In particular, the separation problem for $\overline{\mathbf{H}}$-languages is decidable for any decidable variety of finite groups $\mathbf{H}$. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.