A lattice order on the complex field

Abstract

Birkhoff and Pierce asked in 1956 whether the field $\mathbb{C}$ of complex numbers admits a lattice order compatible with addition and multiplication. We answer the question affirmatively. Using a valuation with value group $\mathbb{Q}$, we construct a positive cone $P$ by a transfinite recursion of countable ordinal length. At each stage we extend a lattice order on a subspace while preserving its lattice operations in the whole field and maintaining multiplicative closure of the ambient positive cone. These subspaces eventually exhaust $\mathbb{C}$. The construction depends on choices of a valuation extension, compatible rational powers and vector-space bases. The cone $P$ contains $1$ and satisfies $\{x\in P\setminus\{0\}:x^{-1}\in P\}=\mathbb{Q}_{>0}$. The induced order is not total: $i$ is incomparable with $0$. For a chosen transcendental element $t$ and $i^2=-1$, one may prescribe $0\vee i=(t+i)/2$ and $0\vee(-i)=(t-i)/2$. Varying the first extension yields exactly $2^{\mathbf c}$ distinct lattice positive cones on $\mathbb{C}$, where $\mathbf c=|\mathbb{C}|$. The count concerns cones as subsets of $\mathbb{C}$.