Yichuan Yang (Beihang University, Beijing)

The complex number field can be directed partially ordered.

Abstract:

The absence of a compatible partial order on the complex number field sets a full stop for many important methods of real analysis. In fact, by Artin-Schreier theory (1926) on formal real fields, it is known that there does not exist a linear order as in the case of the real numbers. Birkhoff and Pierce (1956) asked whether there is a compatible lattice order (which is much weaker than a linear one) on the complex field. Despite of considerable effort, this question is still open. In many situations, however, the existence of a directed order is enough to discuss some important questions. For the complex field, I dealt with that question in J. Alg., 2006. The complete answer is given in J. Alg., 2011 (joint with N. Schwartz). More recently, for any field, we can determine whether it can be directed partially ordered or not.