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Directed Partial Orders on Complex Numbers and Quaternions over Non-Archimedean Linearly Ordered Fields

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Abstract

Let F be a non-archimedean linearly ordered field, and C and H be the field of complex numbers and the division algebra of quaternions over F, respectively. In this paper, a class of directed partial orders on C are constructed directly and concretely using additive subgroup of F +. This class of directed partial orders includes those given in Rump and Wang (J. Algebra 400, 1–7, 2014), and Yang (J. Algebra 295(2), 452–457, 2006) as special cases and we conjecture that it covers all directed partial orders on C such that 1 > 0. It turns out that this construction also works very well on H. We note that none of these directed partial orders is a lattice order on C or H.

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References

  1. Birkhoff, G., Pierce, R.S.: Lattice-ordered rings. An. Acad. Brasil. Ci 28, 41–69 (1956)

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  2. Rump,W., Yang, Y.: Non-archimedean directed fields K(i) with o-subfield and i 2 = −1. J. Algebra 400, 1–7 (2014)

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  3. Yang, Y.: On the existence of directed rings and algebras with negative squares. J. Algebra 295(2), 452–457 (2006)

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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. Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240, Shanghai, 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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Corresponding author

Correspondence to Yuehui Zhang.

Additional information

The third author is supported by NSFC under Grant No. 11271257, by NSF of Shanghai Municipal under Grant No. 13ZR1422500, by Doctorate Program of Education Committee of China under Grant No. 20120073110058

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Ma, J., Wu, L. & Zhang, Y. Directed Partial Orders on Complex Numbers and Quaternions over Non-Archimedean Linearly Ordered Fields. Order 34, 37–44 (2017). https://doi.org/10.1007/s11083-016-9387-y

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  • DOI: https://doi.org/10.1007/s11083-016-9387-y

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