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

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][453][454][455][456][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 .