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arXiv · 2609.20494

Directed partial orders on the complex number field

Wenyi Wang·Ruisong Yuan·Yuehui Zhang·Yuhang Zhu
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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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Wenyi Wang, Ruisong Yuan, Yuehui Zhang, Yuhang Zhu. 2026-09-17. Directed partial orders on the complex number field. https://arxiv.org/abs/2609.20494

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