On semidirectly closed pseudovarieties of aperiodic semigroups

The aim of this work is to study the unknown intervals of the lattice of aperiodic pseudovarieties which are semidirectly closed and answer questions proposed by Almeida in his book “Finite Semigroups and Universal Algebra”. The main results state that the intervals [V*(B2),ER∩LR] and [V*(B21),ER∩A] are not trivial, and that both contain a chain isomorphic to the chain of real numbers. These results are a consequence of the study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B2.