Product varieties of m-groups

Abstract

A new concept of mimicking is introduced. We point out representations that mimic a variety $$ \mathcal{A} $$ of Abelian m-groups and a variety $$ \mathcal{J} $$ of m-groups defined by an identity x* = x-1. It is proved that if a variety $$ \mathcal{U} $$ of m-groups is generated by some class of m-groups, and a variety $$ \mathcal{V} $$ of m-groups is mimicked by some class of m-groups, then their product $$ \mathcal{U}\cdot \mathcal{V} $$ is generated by wreath products of groups in the respective classes. For every natural n, we construct m-groups generating varieties $$ {{\mathcal{J}}^n}=\left( {{{\mathcal{J}}^{n-1 }}} \right)\cdot \mathcal{J} $$ and $$ {{\mathcal{A}}^n}=\left( {{{\mathcal{A}}^{n-1 }}} \right)\cdot \mathcal{A} $$ .