Nonexistence of a compatible lattice order on the complex field

Prove that the complex number field admits no lattice order compatible with its ring operations, thereby establishing a negative answer to the Birkhoff–Pierce problem.

Background

The paper constructs explicit directed partial orders on the complex number field using positive cones derived from integral closures of local rings. It proves that none of the orders in this constructed family is a lattice order: specifically, the elements 0 and i have no least upper bound under any of these orders.

The authors then formulate the broader Birkhoff–Pierce question of whether any lattice order compatible with the ring operations can exist on the complex field. The stated conjecture goes beyond the paper’s results, since the non-lattice proposition rules out lattice orders only within the constructed family.