A simple system of axioms for spaces of signatures
type: Item
instance of: scholarly article
title: A simple system of axioms for spaces of signatures
zbMATH Open document ID: 0668.10027
DOI: 10.1016/0022-4049(89)90112-6
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: The author develops a new system of axioms for spaces of signatures, which is easier to check than the original one. Let (G,\(\cdot)\) be an abelian group of finite (even) exponent and X be a non-empty subset of the character group \(\chi\) (G). The pair (X,G) is said to be a space of signatures if it satisfies certain axioms \((S_ 0)-(S_ 4)\). The author introduces the following new \(axioms\) (S\({}^*_ 4) \) \(\cup_{s\in D<1,x>}D<y,s>=\cup_{t\in D<1,y>}D<x,t>\) for any \(x,y\in G.\) (S\({}^*_ 5) \) For any \(\sigma\in \chi (G)\) with \(\sigma (-1)=-1\) and \(1+\sigma (x)=\sigma (y)+\sigma (z)\) for all x,y,z\(\in G\) such that \(<1,x>\cong <y,z>\), holds \(\sigma\in X.\) Theorem 1 states that (X,G) is a space of signatures iff the axioms \(S_ 2,S_ 3,S^*_ 4\) and \(S^*_ 5\) hold. This has the following application to preorderings of division rings: Let K be a formally real division ring and \(T\subset K\) a preordering. Then the set \(X_ T\) of orderings of K containing T can be viewed as a subset of the character group \(\chi (K^*/T^*)\). Theorem 2 states that \((X_ T,K^*/T^*)\) is a space of signatures (which means a space of orderings in the sense of the author [Trans. Am. Math. Soc. 258, 505-521 (1980; Zbl 0427.10015)]).
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 12E15
zbMATH DE Number: 4093565
zbMATH Keywords: preorderings of higher level
zbMATH Keywords: formally real skew-fields
zbMATH Keywords: new system of axioms
zbMATH Keywords: spaces of signatures
zbMATH Keywords: preorderings of division rings
zbMATH Keywords: formally real division ring
zbMATH Keywords: character group
zbMATH Keywords: space of orderings
Wikidata QID: Q114685096
author: Murray Marshall
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90112-6
OpenAlex ID: W2113381640
cites work: Reduced forms and reduced Witt rings of higher level
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description: scientific article; zbMATH DE number 4093565