I would like to compare partially ordered group $(G,\cdot,\leq)$ and semifield $(S,+,\cdot)$. Here I want to find out the main differences between them. The similarities are clear for me. I think one of the main differences is the total weak order $\leq$. That means, the order $\leq$ is directly connected to the partially ordered group, but not for the semifield. Please any help?
Thanks in advance!