PreprintCC BY 4.0arXiv
The existence of the compatible lattice order on the complex field ℂ
Abstract
In this paper, we give an affirmative answer to the open question posed by G.Birkhoff and R.S.Pierce in 1956 concerning the existence of a compatible lattice order on the complex field ℂ. Using valuation rings theory, basis extension in infinite-dimensional vector spaces and extension of field embeddings to algebraic extensions, we obtain that for every real t>1 transcendental over ℚ, the compatible lattice order can be chosen such that 0≺1, 0∨ i=t, and 1∧ t=0. Consequently, ℂ admits at least 2ℵ0 distinct compatible lattice orders, each extending the usual order on ℚ.
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BibTeX
@misc{he2026existence,
title = {{The existence of the compatible lattice order on the complex field $\mathbbC$}},
author = {Zhengmao He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37155},
eprint = {2610.37155},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37155},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The existence of the compatible lattice order on the complex field ℂ
AU - Zhengmao He
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37155
UR - https://arxiv.org/abs/2610.37155
AB - In this paper, we give an affirmative answer to the open question posed by G.Birkhoff and R.S.Pierce in 1956 concerning the existence of a compatible lattice order on the complex field ℂ. Using valuation rings theory, basis extension in infinite-dimensional vector spaces and extension of field embeddings to algebraic extensions, we obtain that for every real t>1 transcendental over ℚ, the compatible lattice order can be chosen such that 0≺1, 0∨ i=t, and 1∧ t=0. Consequently, ℂ admits at least 2^ℵ_0 distinct compatible lattice orders, each extending the usual order on ℚ.
ER -