  Abelian groups having partial orderings compatible with their binary
operations have long been studied in the literature. In particular,
lattice-ordered abelian groups constitute a universal-algebraic variety, and
thus form a category which is monadic over the category of sets. The current
paper studies the more general case of quasi-ordered abelian groups,
identifying some of their more fundamental properties and their relationships
to partially ordered and lattice-ordered groups. We reinterpret the category of
quasi-ordered abelian groups with order preserving morphisms by examining the
interplay between the group and the set of all positive elements under the
quasi-ordering. The main result shows that the category of quasi-ordered
abelian groups is monadic over the category of set monomorphisms.
