Consider a “pseudo-ordering” on the complex field defined by saying a complex number is positive if and only if is positive. Which of the three order axioms (listed in the previous exercise) are satisfied.
The second and third axioms are satisfied, but not the first.
The second axiom states that for every with either is positive or is positive, but not both. By our definition, positive implies . Therefore, since we have . Hence, is not positive.
The third axiom is satisfied since by our definition of positive if and , then is not positive.
Finally, the first axiom fails since if we take , then is positive, but
is not positive.