  In this article we explore a non-abelian torsion theory in the category of
preordered groups: the objects of its torsion-free subcategory are the
partially ordered groups, whereas the objects of the torsion subcategory are
groups (with the total order). The reflector from the category of preordered
groups to this torsion-free subcategory has stable units, and we prove that it
induces a monotone-light factorization system. We describe the coverings
relative to the Galois structure naturally associated with this reflector, and
explain how these coverings can be classified as internal actions of a Galois
groupoid. Finally, we prove that in the category of preordered groups there is
also a pretorsion theory, whose torsion subcategory can be identified with a
category of internal groups. This latter is precisely the subcategory of
protomodular objects in the category of preordered groups, as recently
discovered by Clementino, Martins-Ferreira, and Montoli.
