I am trying to find a field (I'll settle for other stuff - some type of ring) that is ordered and $a>0,\,b>0$ implies $a+b > 0$ yet $a,b > 0$ doesn't imply $ab > 0$.
Much of the problem seems to be working to satisfy the distributive law.
Cheers.
edit: and $a>0$ or $a=0$ or $a<0$ holds for all $a$ and only one relation does in-fact hold.