A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. There are found necessary and sufficient conditions under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.