We compare the classical semidirect product V * W of pseudovarieties of semigroups with the recently found 'regular' semidirect product V * W of e-pseudovarieties of regular semigroups. Applications of our findings include a substantial class of additions to he 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 Kadourek.
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