Publication details

On a locality-like property of the pseudovariety J



Year of publication 2018
Type Article in Periodical
Magazine / Source Periodica mathematica Hungarica
MU Faculty or unit

Faculty of Science

Keywords Pseudovarieties of finite monoids and finite categories; Locally finite varieties of monoids and categories; Finitely generated relatively free monoids and categories; Semidirect products of pseudovarieties of finite monoids
Description It is well known that the pseudovariety J of all J-trivial monoids is not local, which means that the pseudovariety gJ of categories generated by J is a proper subpseudovariety of the pseudovariety lJ of categories all of whose local monoids belong to J. In this paper, it is proved that the pseudovariety J enjoys the following weaker property. For every prime number p, the pseudovariety lJ is a subpseudovariety of the pseudovariety g(J*Abp), where Abp is the pseudovariety of all elementary abelian p-groups and J*Abp is the pseudovariety of monoids generated by the class of all semidirect products of monoids from J by groups from Abp. As an application, a new proof of the celebrated equality of pseudovarieties PG=BG is obtained, where PG is the pseudovariety of monoids generated by the class of all power monoids of groups and BG is the pseudovariety of all block groups.

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