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Positivity

pp 1–7 | Cite as

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

  • Jingjing Ma
  • Liusan Wu
  • Yuehui ZhangEmail author
Article
  • 7 Downloads

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.

Keywords

Non-archimedean linearly ordered field Directed partially ordered algebra Directed partial order Special convex subset Lattice order 

Mathematics Subject Classification

Primary 06F25 Secondary 20M25 

Notes

References

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Houston-Clear LakeHoustonUSA
  2. 2.College of EngineeringNanjing Agricultural UniversityNanjingChina
  3. 3.School of Mathematical Sciences, MOE-LSCShanghai Jiao Tong UniversityShanghaiChina

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