Preorders, left orderings, and valuation of near-fields
The concept of a near-field differs from the one of a skew-field just by the lack of one of the distributive laws. All finite near-fields were classified in a monumental paper by \textit{H. Zassenhaus} [Abh. Math. Semin. Hamb. Univ. 11, 187--220 (1935; Zbl 0011.10302 and JFM 61.0126.01)]. In this paper, (left) ordered near-fields \(F\) are studied; these are near-fields with a linear order relation \(\leq\) such that \(x \leq y\) implies \(a+x \leq a+y\) for each \(a\) and \(bx \leq by\) for each positive \(b\). Using the positive cone \(P\), this means that both \(P+P\) and \(P.P\) are contained in \(P\). The near-field \(F\) is called formally left-real if \(-1\) is not a sum of products of squares in \(F\). The analogue of Artin-Schreier's theorem says that \(F\) is formally left real iff \(F\) admits a left ordering. A map \(v\) from \(F\) to a linearly ordered set \((\Gamma, \leq)\) with smallest element \(0\) is called a valuation of \(F\) if \(v(x)=0\) holds iff \(x=0\), \(v(x+y) \leq \max(v(x),v(y))\) and if \(v(x) \leq v(y)\) implies \(v(ax) \leq v(ay)\) for all \(x,y,a \in F\). The paper studies the connections between left orderings \(P\) and valuations \(v\) which are compatible with \(P\) (i.e., where \(0 \leq x \leq y\) always implies \(v(x) \leq v(y)\)). The results are too technical to be given here.
scientific article; zbMATH DE number 5911448
scientific article