Preorders, left orderings, and valuation of near-fields
type: Item
instance of: scholarly article
title: Preorders, left orderings, and valuation of near-fields
zbMATH Open document ID: 1231.12004
author: Christian Karpfinger
published in: Results in Mathematics
publication date: 2011-06-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 5911448
zbMATH Keywords: near-field
zbMATH Keywords: left order
zbMATH Keywords: valuation
zbMATH Keywords: positive cone
zbMATH Keywords: formally real
MaRDI profile type: Publication
full work available at URL: s00025-011-0098-9
OpenAlex ID: W2056954322
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DOI: 10.1007/S00025-011-0098-9
description: scientific article; zbMATH DE number 5911448