On finitely based pseudovarieties of the forms \({\mathbf V}*{\mathbf D}\) and \({\mathbf V}*{\mathbf D}_n\)
This paper studies how the operators \(*\mathbb{D}\) and \(*\mathbb{D}_n\) on the lattice of pseudovarieties behave with respect to the finite basis property. The principal result shows that if \(\mathbb{V}\) is a pseudovariety of semigroups whose global is finitely based, then the aforementioned operators preserve the finite basis property. This is proved by a refinement of the basis theorem for semidirect products of \textit{J. Almeida} and \textit{P. Weil} [J. Pure Appl. Algebra 123, No. 1-3, 1-50 (1998; Zbl 0891.20037)] to the case of \(\mathbb{D}\). Using consequences of Tilson's delay theorem, they show that, for a monoidal pseudovariety \(\mathbb{V}\), the finite basis problems for the global of \(\mathbb{V}\), \(\mathbb{V}*\mathbb{D}\), and \(\mathbb{V}*\mathbb{D}_n\) are equivalent. Perhaps the most important part of the paper is a construction used by the authors to show that if \(\mathbb{V}\) contains the 5-element aperiodic Brandt monoid \(B_2\), then one can ``effectively'' construct a basis of pseudoidentities for the global of \(\mathbb{V}\). This construction, which makes use of Reilly's characterization of the variety of inverse semigroups generated by \(B_2\), shows that such pseudovarieties are finitely based if and only if their globals are. This construction is used in a later paper by Almeida and Steinberg on the Rhodes complexity problem.
scientific article; zbMATH DE number 1398843