Varieties of exponential $MR$-groups

In this paper we introduce the notion of a variety of exponential $MR$-groups and tensor completions of groups in varieties. We study relationships between free groups of a given variety under different rings of scalars and describe varieties of abelian $MR$-groups. Moreover, in the category of $MR$-groups, we consider several analogs of $n$-class nilpotent groups. We got that the completion of a 2-class nilpotent group is a 2-class nilpotent.