A lattice order on the complex field

Abstract

Birkhoff and Pierce asked in 1956 whether the field ℂ of complex numbers admits a lattice order compatible with addition and multiplication. We answer the question affirmatively. Using a valuation with value group 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 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∈ P∖{0}:x⁻¹∈ P}=ℚ_>0. The induced order is not total: i is incomparable with $0$. For a chosen transcendental element t and i²=−1, one may prescribe 0∨ i=(t+i)/2 and 0∨(−i)=(t−i)/2. Varying the first extension yields exactly 2^c distinct lattice positive cones on ℂ, where c=|ℂ|. The count concerns cones as subsets of ℂ.

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BibTeX RIS

BibTeX

@misc{bao2026lattice,
  title = {{A lattice order on the complex field}},
  author = {Hanzhen Bao and Yuehui Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38300},
  eprint = {2610.38300},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38300},
}

RIS

TY  - UNPB
TI  - A lattice order on the complex field
AU  - Hanzhen Bao
AU  - Yuehui Zhang
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38300
UR  - https://arxiv.org/abs/2610.38300
AB  - Birkhoff and Pierce asked in 1956 whether the field ℂ of complex numbers admits a lattice order compatible with addition and multiplication. We answer the question affirmatively. Using a valuation with value group 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 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∈ P∖{0}:x^−1∈ P}=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∨ i=(t+i)/2 and 0∨(−i)=(t−i)/2. Varying the first extension yields exactly 2^c distinct lattice positive cones on ℂ, where c=|ℂ|. The count concerns cones as subsets of ℂ.
ER  -

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