  For a commutative, unital and integral quantale V, we generalize to V-groups
the results developed by Gran and Michel for preordered groups. We first of all
show that, in the category V-Grp of V-groups, there exists a torsion theory
whose torsion and torsion-free subcategories are given by those of indiscrete
and separated V-groups, respectively. It turns out that this torsion theory
induces a monotone-light factorization system that we characterize, and it is
then possible to describe the coverings in V-Grp. We next classify these
coverings as internal actions of a Galois groupoid. Finally, we observe that
the subcategory of separated V-groups is also a torsion-free subcategory for a
pretorsion theory whose torsion subcategory is the one of symmetric V-groups.
As recently proved by Clementino and Montoli, this latter category is actually
not only coreflective, as it is the case for any torsion subcategory, but also
reflective.
