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Directed partial orders on the field of generalized complex numbers with \(1\not >0\)

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Abstract

Let F be a non-archimedean o-field and \(C=F(i)\) be the field of generalized complex numbers over F. In Ma et al. (Order 35:461–466, 2018), all directed partial orders on C with \(1>0\) are classified using admissible semigroups of \(F^{+}\). This paper classifies all the directed partial orders on C with \(1\not >0\) using special convex subsets of \(F^{+}\). As a consequence, the (non-archimedean) analogue of Fuchs’ question in 1963 is answered completely.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston-Clear Lake, 2700 Bay Area Boulevard, Houston, TX, 77058, USA

    Jingjing Ma

  2. College of Engineering, Nanjing Agricultural University, Nanjing, 210095, China

    Liusan Wu

  3. School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yuehui Zhang

Authors
  1. Jingjing Ma
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  2. Liusan Wu
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  3. Yuehui Zhang
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Correspondence to Yuehui Zhang.

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The third author is supported by NSFC under Grant Nos. 11671258 and 11771280, by NSF of Shanghai Municipal under Grant No. 17ZR1415400.

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Ma, J., Wu, L. & Zhang, Y. Directed partial orders on the field of generalized complex numbers with \(1\not >0\). Positivity 23, 1001–1007 (2019). https://doi.org/10.1007/s11117-019-00647-7

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  • DOI: https://doi.org/10.1007/s11117-019-00647-7

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