Pseudovarieties, regular semigroups, and semidirect products

We compare the classical semidirect product V * W of pseudovarieties of semigroups with the recently found ' regular' semidirect product Mathematical script capital V * e Script W sign of e-pseudovarieties of regular semigroups. Applications of our findings include a substantial class of additions to the list of pseudovarieties generated by classes of regular semigroups and for which the membership problem is solved. For example, we compute the pseudovarieties generated by all locally inverse and locally orthodox semigroups, respectively, thereby answering a question raised by Almeida and Kaďourek.