Lattice-ordered reduced special groups

Annals of Pure and Applied Logic 132 (1):27-49 (2005)

Abstract
Special groups [M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, Memoirs Amer. Math. Soc., vol. 689, Amer. Math. Soc., Providence, RI, 2000] are a first-order axiomatization of the theory of quadratic forms. In Section 2 we investigate reduced special groups which are a lattice under their natural representation partial order ; we show that this lattice property is preserved under most of the standard constructions on RSGs; in particular finite RSGs and RSGs of finite chain length are lattice ordered. We prove that the lattice property fails for the RSGs of function fields of real algebraic varieties over a uniquely ordered field dense in its real closure, unless their stability index is 1 . We show that Open Problem 1 has a positive answer for the RSG of the field Q . In the final section we explore the meaning of Open Problem 1 for formally real fields, in terms of their orders and real valuations; we introduce the notion of “parameter-rank” of a positive-primitive first-order formula of the language for special groups
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DOI 10.1016/j.apal.2004.06.005
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Open Questions in the Theory of Spaces of Orderings.Murray A. Marshall - 2002 - Journal of Symbolic Logic 67 (1):341-352.
Lattice Theory.Garrett Birkhoff - 1940 - Journal of Symbolic Logic 5 (4):155-157.

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