The pseudoidentity problem and reducibility for completely regular semigroups (Q2730741)

For a pseudovariety \(\mathbf H\) of groups, let \(\overline{\mathbf H}\) be the pseudovariety consisting of all finite semigroups all of whose subgroups lie in \(\mathbf H\). The main result of the article (Theorem 4.1) gives necessary and sufficient conditions for a pseudidentity \(u=v\) to hold in \(\mathbf{CR}\cap\overline{\mathbf H}\) where \(\mathbf{CR}\) is the pseudovariety of all finite completely regular semigroups and \(\mathbf H\) is a given pseudovariety of groups. Let \(\kappa\) be the signature consisting of the usual semigroup multiplication and the standard unary implicit operation \(\omega\) (for any element \(x\) of a semigroup \(S\), \(x^\omega\) is the only idempotent power of \(x\)). The authors propose a strengthened version of Ash's inevitability theorem (\(\kappa\)-reducibility of the pseudovariety \(\mathbf G\) of all finite groups) as an open problem and show that if this stronger version holds then \(\mathbf{CR}\) is \(\kappa\)-reducible and, therefore, hyperdecidable (Theorem 6.4).