On semidirectly closed pseudovarieties of aperiodic semigroups

Abstract

The aim of this work is to study the unknown intervals of the lattice of aperiodic pseu\-do\-va\-rie\-ties which are semidirectly closed and answer questions proposed by J.~Almeida in his book ``Finite Semigroups and Universal Algebra". The main results state that the intervals $[\mathbf{V}^*(B_2),\mathbf{ER}\cap\mathbf{LR}]$ and $[\mathbf{V}^*(B^1_2),\mathbf{ER}\cap\mathbf{A}]$ 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 $B_2$.