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