On the equation V*G=EV

This note uses some recent powerful tools related with semidirect products V * W of pseudovarieties of semigroups, particularly when the second factor is the pseudovariety G of all finite groups, to give syntactic proofs of equalities of the form V*G=ℰV, where ℰV denotes the pseudovariety consisting of the finite semigroups whose idempotents generate subsemigroups from V. Subpseudovarieties V of script D sign S are considered, including all subpseudovarieties of ℒI, script D sign A, script D sign S itself, and J, the latter providing a new proof of a result of Henckell and Rhodes. © 2002 Elsevier Science B.V. All rights reserved.