Complete kappa-reducibility of pseudovarieties of the form DRH

TL;DR

This paper proves that the property of complete kappa-reducibility in pseudovarieties of groups H extends to the associated pseudovariety DRH of finite semigroups, establishing a bidirectional equivalence.

Contribution

It generalizes the concept of complete kappa-reducibility from groups to semigroups of the form DRH, using adapted tools from previous work.

Findings

Complete kappa-reducibility of DRH when H is completely kappa-reducible.

The converse: if DRH is completely kappa-reducible, then H is.

Extension of techniques from R-trivial monoids to DRH setting.

Abstract

We denote by kappa the implicit signature that contains the multiplication and the (omega-1)-power. It is proved that for any completely kappa-reducible pseudovariety of groups H, the pseudovariety DRH of all finite semigroups whose regular R-classes are groups in H is completely kappa-reducible as well. The converse also holds. The tools used by Almeida, Costa, and Zeitoun for proving that the pseudovariety of all finite R-trivial monoids is completely kappa-reducible are adapted for the general setting of a pseudovariety of the form DRH.