On a locality-like property of the pseudovariety $$\mathbf {J}$$ J

Abstract

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.