Spaces of orderings and Witt rings of planar ternary rings
type: Item
instance of: scholarly article
title: Spaces of orderings and Witt rings of planar ternary rings
zbMATH Open document ID: 0679.10017
DOI: 10.1016/0022-4049(89)90156-4
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: The author translates part of the theory of ordered fields into the context of ordered projective planes which enables him to apply Marshall's theory of spaces of orderings [\textit{M. Marshall}, Can. J. Math. 31, 320-330 (1979; Zbl 0412.10012)]. As a main result, he proves that the set of orderings of an arbitrary orderable planar ternary ring (thus the set of orderings of an arbitrary orderable projective or affine plane) forms a space of orderings in the sense of Marshall. The first section contains the basic definitions and results. The author describes the subsets H(T) and Q(T) which are the equivalents to \(K^{*2}\) and resp. \(\Sigma K^{*2}\) in the theory of ordered fields K. A planar ternary ring (shortly PTR) T is called formally-real iff 0 is not a finite sum of elements of H(T) and equivalently iff T is orderable (1.2). Then he introduces preorderings and orderings using the standard axioms. The orderings correspond 1-1 to the orders which are certain quadratic characters of \(T^*\) fulfilling axioms similar to the ones of signature functions in formally-real fields (1.3). This leads to the equivalence between ``formally-real'' and the existence of a preordering on T (which always extends to an ordering). These results allow the author to bring PTR's into the context of Marshall's abstract theory of forms. The main result of section 2 is Theorem 2.3: If T is a formally-real PTR and P a preordering of T, then \((X/P,T^*/P)\) is a space of orderings. The proof makes use of the simplified set of axioms for spaces of signatures (resp. orderings) which were introduced by \textit{M. Marshall} [J. Pure Appl. Algebra 57, No.2, 159-164 (1989; Zbl 0668.10027)]. Theorem 2.5 states that this does not yield any ``new'' finite spaces of orderings, as any finite space of orderings of a formally-real PTR already is isomorphic to the space of orderings of some Pythagorean commutative field. Section 3 deals with the abstract (reduced) Witt rings of PTR's. The author first…
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12K99
zbMATH DE Number: 4112688
zbMATH Keywords: ordered projective planes
zbMATH Keywords: spaces of orderings
zbMATH Keywords: set of orderings
zbMATH Keywords: orderable planar ternary ring
zbMATH Keywords: Marshall's abstract theory of forms
zbMATH Keywords: Witt ring
zbMATH Keywords: reduced Witt ring
author: Franz B. Kalhoff
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90156-4
OpenAlex ID: W1999653830
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description: scientific article; zbMATH DE number 4112688