Directed partial orders on complex numbers and quaternions. II (Q346410)

Let \((R,\leq)\) be an ordered ring. Its positive cone is the subset \(R^+=\{x\in R:x\geq 0\}\). Then \((R,\leq)\) is a directed ring if each element of \(R\) is the difference of two elements of \(R^+\). Let \(F\) be a partially ordered field with a directed partial order and \(K\) a non-Archimedean totally ordered subfield such that \(F^+\cap K=K^+\). Let \(C_K=K+Ki\) be the field of complex numbers over \(K\) where \(i^2=-1\) and \(H_K=K+Ki+Kj+Kk\) be the division algebra of quaternions over \(K\). In this paper, the author constructs directed partial orders for \(C_K\) and \(H_K\). It is also shown that \(H_{\mathbb{R}}\) cannot be a directed algebra if the field \(\mathbb{R}\) of real numbers is equipped with the usual total order.