Aperiodic Pointlikes and Beyond

TL;DR

This paper proves the decidability of pointlike sets for certain semigroup pseudovarieties, simplifying previous proofs and extending results to a broader class of semigroups.

Contribution

It introduces a simpler proof for aperiodic semigroups and generalizes the decidability of pointlike sets to semigroups with subgroup structures based on recursive prime sets.

Findings

Decidability of pointlike sets for semigroups with specific subgroup structures

Simplified proof for aperiodic semigroups

Extension of Henckell's result to broader classes

Abstract

We prove that if $\pi$ is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are $\pi$-groups. In particular, when $\pi$ is the empty set, we obtain Henckell's decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof.