Directed partial orders on complex numbers and quaternions
Jingjing Ma · Journal of Algebra and Its Applications · 2024
Let [Formula: see text] be an integral domain that is algebraic over [Formula: see text]. It is shown that each directed maximal partial order on [Formula: see text] is an Archimedean total order. Let [Formula: see text] be a subfield of [Formula: see text] and [Formula: see text] be the complex field over [Formula: see text]. As a consequence of the above result, if [Formula: see text] is algebraic over [Formula: see text], then [Formula: see text] does not have a directed partial order making it a partially ordered ring. In particular, [Formula: see text] cannot be a lattice-ordered ring. The result is proved for certain partially ordered algebras of quaternions as well.