\(\mathsf E\)-local pseudovarieties.
The author says that a pseudovariety \(\mathbf V\) of finite semigroups is \(\mathsf E\)-local if, for every finite semigroup \(S\), the subsemigroup generated by all idempotents of \(S\) lies in \(\mathbf V\) whenever, for each regular \(\mathcal D\)-class \(D\) of \(S\), the subsemigroup generated by all idempotents of \(D\) lies in \(\mathbf V\). She exhibits several examples of \(\mathsf E\)-local pseudovarieties; in particular, she observes that for every pseudovariety \(\mathbf H\) of finite groups, the class of all finite semigroups whose subgroups lie in \(\mathbf H\) forms an \(\mathsf E\)-local pseudovariety. Further, the author characterizes \(\mathsf E\)-local pseudovarieties that consist of finite semigroups in whose idempotent-generated subsemigroups each regular \(\mathcal D\)-class forms a subsemigroup. She also provides several sufficient conditions for a pseudoidentity to define an \(\mathsf E\)-local pseudovariety.
scientific article; zbMATH DE number 6103697