I’m a bit out of my depth here. I understood an “ordered group” as a group with an order on its elements. That clearly can be finite. If it’s more than that the question would be why we should assume whatever further axioms characterize it.
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.
I’m a bit out of my depth here. I understood an “ordered group” as a group with an order on its elements. That clearly can be finite. If it’s more than that the question would be why we should assume whatever further axioms characterize it.
from wikipedia:
So if a > 0, a+a > a etc. which results means the group has to be torsion free.