Profinite categories and semidirect products
This very substantial paper is motivated by a complementary question pair -- for a given semidirect product \({\mathbf V}*{\mathbf W}\) of pseudovarieties of semigroups when is a given pseudoidentity valid in \({\mathbf V}*{\mathbf W}\) and what is a basis of pseudoidentities for \({\mathbf V}*{\mathbf W}\)? Although the authors go through considerable length to introduce various facets of their theory: implicit operations, pseudovarieties of categories, the notion of a local pseudovariety of monoids, and relational morphisms (which form an ingredient of a certain extension of the idea of semidirect product), the paper is largely an attempt to extend Tilson's Theory of Derived Categories by providing what the authors see as the `missing ingredient': a Reiterman type theorem for pseudovarieties of categories (Theorem 2.7) which admits a short proof based on the projective limit of profinite categories. The authors use their extended semidirect product to produce a description of a basis of pseudo-identities for a given semidirect product which, although not effective, provides membership criteria in many interesting cases. A generalisation of Tilson's Delay Theorem is given along with some consequences and applications: for instance the lattice of pseudovarieties of categories is naturally embedded in the lattice of pseudovarieties of semigroups.
scientific article; zbMATH DE number 1121660
scientific article