Moura, Ana2018-03-212018-03-212012http://hdl.handle.net/10400.22/11162Generalizingapropertyofthepseudovarietyofallaperiodicsemi- groups observed by Tilson, we call E-local a pseudovariety V which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular D-classes. In this paper, we present sev- eral sufficient or necessary conditions for a pseudovariety to be E-local or for a pseudoidentity to define an E-local pseudovariety. We also determine several examples of the smallest E-local pseudovariety containing a given pseudovariety.engPseudovariety DASubsemigroup generated by its idempotentsRegular D-classesRegular D-classesjournal article10.1007/s00233-012-9413-3