Category of $\mathrm{MR}$-groups over a ring $\mathrm R$

The category of exponential $\mathrm{MR}$-groups for an associative ring $\mathrm R$ with unity is defined in [1]. The present paper is devoted to the study of partial exponential $\mathrm{MR}$-groups which are isomorphically embedded in their tensor completion over the ring $\mathrm R$. The key to its understanding is the notion of tensor completion introduced in [1]. As a consequence, the description of free $\mathrm{MR}$-groups in the language of group constructions is obtained.