Tannakian formalism for fiber functors over tensor categories

It is well known that the pseudovariety \(\mathbf {J}\) of all \(\mathscr {J}\)-trivial monoids is not local, which means that the pseudovariety \(g\mathbf {J}\) of categories generated by \(\mathbf {J}\) is a proper subpseudovariety of the pseudovariety \(\ell \mathbf {J}\) of categories all of whose local monoids belong to \(\mathbf {J}\). In this paper, it is proved that the pseudovariety \(\mathbf {J}\) enjoys the following weaker property. For every prime number p, the pseudovariety \(\ell \mathbf {J}\) is a subpseudovariety of the pseudovariety \(g(\mathbf {J}*\mathbf {Ab}_p)\), where \(\mathbf {Ab}_p\) is the pseudovariety of all elementary abelian p-groups and \(\mathbf {J}*\mathbf {Ab}_p\) is the pseudovariety of monoids generated by the class of all semidirect products of monoids from \(\mathbf {J}\) by groups from \(\mathbf {Ab}_p\). As an application, a new proof of the celebrated equality of pseudovarieties \(\mathbf {PG}=\mathbf {BG}\) is obtained, where \(\mathbf {PG}\) is the pseudovariety of monoids generated by the class of all power monoids of groups and \(\mathbf {BG}\) is the pseudovariety of all block groups.