Because of the apparent unfeasibility of the classification of finite semigroups up to isomorphism, and also because of the relationship between the so called pseudovarieties of semigroups and certain families of rational languages, which provide a suitable connection with the theory of finite automata, the theory of pseudovarieties has been intensively studied.

In the attempt of finding a method for computing the complexity of a finite semigroup in terms of the structure of its subsemigroups, Tilson observed a property of the pseudovariety of all aperiodic semigroups that we extend to any pseudovariety. We call E-local a pseudovariety V  which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular D-classes. Several techniques were used in the attempt to characterize the E-local pseudovarieties, from the properties of idempotent-generated semigroups and blocks of such subsemigroups to the profinite ones.

In this talk, I will present my contribution towards the complete characterization of E-local pseudovarieties.