Spaces of Orderings and Abstract Real Spectra

1996 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Spaces Of Orderings

scholarly journals Direct limits of finite spaces of orderings

1984 ◽  
Vol 112 (2) ◽  
pp. 391-406 ◽  
Author(s):  
Mieczysław Kula ◽  
Murray Marshall ◽  
Andrzej Sładek
Keyword(s):  
Spaces Of Orderings ◽  
Direct Limits ◽  
Finite Spaces

Quotients and Inverse Limits of Spaces of Orderings

1979 ◽  
Vol 31 (3) ◽  
pp. 604-616 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Sums Of Squares ◽  
Inverse Limits ◽  
Witt Ring ◽  
Spaces Of Orderings ◽  
Image Position

A connection between the theory of quadratic forms defined over a given field F, and the space XF of all orderings of F is developed by A. Pfister in [12]. XF can be viewed as a set of characters acting on the group F×/ΣF×2, where ΣF×2 denotes the subgroup of F× consisting of sums of squares. Namely, each ordering P ∈ XF can be identified with the characterdefined byIt follows from Pfister's result that the Witt ring of F modulo its radical is completely determined by the pair (XF, F×/ΣF×2).

Finite Spaces of Signatures

1989 ◽  
Vol 41 (5) ◽  
pp. 808-829 ◽  
Author(s):  
Victoria Powers
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Character Group ◽  
Witt Rings ◽  
Ordered Field ◽  
Spaces Of Orderings ◽  
Ordered Fields ◽  
Skew Fields ◽  
Finite Spaces ◽  
Field Setting

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

scholarly journals TOPOLOGY ON THE SPACES OF ORDERINGS OF GROUPS

2004 ◽  
Vol 36 (04) ◽  
pp. 519-526 ◽  
Author(s):  
ADAM S. SIKORA
Keyword(s):  
Spaces Of Orderings

scholarly journals Direct limits of finite spaces of orderings

1984 ◽  
Vol 14 (4) ◽  
pp. 929-930
Author(s):  
M. Kula ◽  
M. Marshall ◽  
A. Slader
Keyword(s):  
Spaces Of Orderings ◽  
Direct Limits ◽  
Finite Spaces

Spaces of Orderings

1996 ◽  
pp. 85-123
Author(s):  
Carlos Andradas ◽  
Ludwig Bröcker ◽  
Jesús M. Ruiz
Keyword(s):  
Spaces Of Orderings

THE pp CONJECTURE FOR SPACES OF ORDERINGS OF RATIONAL CONICS

2007 ◽  
Vol 06 (02) ◽  
pp. 245-257 ◽  
Author(s):  
PAWEŁ GŁADKI ◽  
MURRAY MARSHALL
Keyword(s):  
Function Field ◽  
Stability Index ◽  
Rational Numbers ◽  
Rational Points ◽  
Basic Question ◽  
Proper Subspace ◽  
Spaces Of Orderings ◽  
The Stability ◽  
Irreducible Conic

First counterexamples are given to a basic question raised in [10]. The paper considers the space of orderings (X,G) of the function field of a real irreducible conic [Formula: see text] over the field ℚ of rational numbers. It is shown that the pp conjecture fails to hold for such a space of orderings when [Formula: see text] has no rational points. In this case, it is shown that the pp conjecture "almost holds" in the sense that, if a pp formula holds on each finite subspace of (X,G), then it holds on each proper subspace of (X,G). For pp formulas which are product-free and 1-related, the pp conjecture is known to be true, at least if the stability index is finite [11]. The counterexamples constructed here are the simplest sort of pp formulas which are not product-free and 1-related.

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