  We introduce a notion of compatible quasi-ordered groups which unifies valued
and ordered abelian groups. It was proved in a paper by Fakhruddin that a
compatible quasi-order on a field is always either an order or a valuation. We
show here that the group case is more complicated than the field case and
describe the general structure of a compatible quasi-ordered abelian group. We
also develop a notion of quasi-order-minimality and establish a connection with
C-minimality, thus answering a question of F.Delon.
