Pointlike sets for varieties determined by groups

TL;DR

This paper characterizes pointlike sets in finite semigroups relative to varieties of groups, providing an effective method for decidability and generalizing previous results on aperiodic pointlikes.

Contribution

It offers a new characterization of pointlike sets for varieties determined by groups, enabling effective decision procedures for related separation problems.

Findings

Characterization of pointlike sets for varieties of finite semigroups.

Decidability of the separation problem for $ar{f H}$-languages.

Generalization of Henckell's theorem on aperiodic pointlikes.

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.