Aperiodic Pointlikes and Beyond

Abstract

We prove that if $π$ is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are $π$-groups. In particular, when $π$ 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.