PreprintarXiv licencearXiv
Directed partial orders on the complex number field
Abstract
We construct a class of positive cones that make C into a directed partially ordered ring. The positive cones are defined using integral closures of local rings, associated with a transcendence basis and a chosen real generator. A localization criterion also yields such orders on every transcendental extension of Q. For a fixed coefficient field, we prove that two real generators define the same cone if and only if they differ by an affine map with positive real-algebraic slope and translation algebraic over that field. The nonzero positive elements are closed under inversion. None of these orders is a lattice order.
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BibTeX
@misc{wang2026directed,
title = {{Directed partial orders on the complex number field}},
author = {Wenyi Wang and Ruisong Yuan and Yuehui Zhang and Yuhang Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20494},
eprint = {2609.20494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20494},
}
RIS
TY - UNPB
TI - Directed partial orders on the complex number field
AU - Wenyi Wang
AU - Ruisong Yuan
AU - Yuehui Zhang
AU - Yuhang Zhu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20494
UR - https://arxiv.org/abs/2609.20494
AB - We construct a class of positive cones that make C into a directed partially ordered ring. The positive cones are defined using integral closures of local rings, associated with a transcendence basis and a chosen real generator. A localization criterion also yields such orders on every transcendental extension of Q. For a fixed coefficient field, we prove that two real generators define the same cone if and only if they differ by an affine map with positive real-algebraic slope and translation algebraic over that field. The nonzero positive elements are closed under inversion. None of these orders is a lattice order.
ER -