If it's more than that the question would be why we should assume whatever further axioms characterize it

from wikipedia:

a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b

So if a > 0, a+a > a etc. which results means the group has to be torsion free.