On the existence of non-negative bases in subgroups of free groups of Schreier varieties

We show that a subgroup $H$ of a free group $F(X)$ has a non-negative (with respect to $X$) basis if and only if $H$ is generated by the set of all its non-negative (with respect to $X$) elements. A similar result is proved for subgroups of free Abelian groups.