Join decompositions of pseudovarieties involving semigroups with commuting idempotents (Q1602666)

Denote by \(\mathbf{DH}\) the pseudovariety of all finite semigroups whose regular \(\mathcal J\)-classes are groups from a given pseudovariety \(\mathbf H\) of groups; by \(\mathbf J\) the pseudovariety of all finite \(\mathcal J\)-trivial semigroups; by \(\mathbf{ECom}\) the pseudovariety of all finite semigroups in which the idempotents commute; and by \(\mathbf{Ab}\) the pseudovariety of all finite Abelian groups. The pseudovariety \(\mathbf H\) of groups is said to be arborescent if \(({\mathbf H}\cap\mathbf{Ab})*{\mathbf H}={\mathbf H}\), where \(*\) denotes semidirect product. The equation \(\mathbf{DH}\cap\mathbf{ECom}=({\mathbf J}\cap\mathbf{ECom})\lor{\mathbf H}\) was established for arborescent pseudovarieties of groups \(\mathbf H\) by \textit{P. Weil} and the reviewer [Int. J. Algebra Comput. 4, No. 3, 375-403 (1994; Zbl 0818.20024)]. The original proof of this join decomposition is non-constructive in the sense that, given a semigroup \(S\) from \(\mathbf{DH}\cap\mathbf{ECom}\) with \(n\) elements, it was not even known whether there was any primitive recursive function \(f(n)\) such that \(S\) divides \(T\times U\) for some \(T\in{\mathbf J}\cap\mathbf{ECom}\) and \(U\in{\mathbf H}\) with cardinalities \(|T|\) and \(|U|\) not exceeding \(f(n)\). In the present paper, the author gives a constructive proof of the same equality which applies for a larger set of pseudovarieties of groups. With the above notation, the proof exhibits the required semigroups \(T\) and \(U\).