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Directed partial orders over non-archimedean o-fields

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Abstract

Let F be a non-archimedean o-field, \(C=F(i)\) the imaginary quadratic extension field of F with \(i^2=-1\). In this paper, all directed partial orders on C are classified via the new concept of doubly convex set consisting of some infinitesimals. This unifies the previous work Ma et al. (Order 34(1):37–44, 2017; Order 35(3):461–466, 2018; Positivity 24(3):1001–1007, 2019). It is surprising that this new theory applies well to the quaternions \(H=F+Fi+Fj+Fk\) over F and all directed partial orders on H are classified. As an application, the Fuchs’ problem is answered negatively for H.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningxia University, 489 Helanshan West Road, Yinchuan, 750021, China

    Yuehui Zhang

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Zhipeng Xu & Yuehui Zhang

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  1. Zhipeng Xu
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  2. Yuehui Zhang
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Correspondence to Yuehui Zhang.

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Supported by NSFC under Grant Nos. 11771280 and 11671258, by NSF of Shanghai Municipal under Grant No. 17ZR1415400.

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Xu, Z., Zhang, Y. Directed partial orders over non-archimedean o-fields. Positivity 24, 1279–1291 (2020). https://doi.org/10.1007/s11117-019-00732-x

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  • DOI: https://doi.org/10.1007/s11117-019-00732-x

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