Lattice-ordered reduced special groups

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 (RSG) which are a lattice under their natural representation partial order (for motivation see Open Problem 1, Introduction); 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 (Section 3). We show that Open Problem 1 (a strong local-global principle) has a positive answer for the RSG of the field Q(X) (Theorem 4.1). 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 (and employ) the notion of “parameter-rank” of a positive-primitive first-order formula of the language for special groups. © 2004 Elsevier B.V. All rights reserved. ∗ Corresponding author at: Equipe de Logique Mathématique, UMR 7056, 2, Place Jussieu Case 7012, 75251 Paris Cedex 05, France. E-mail addresses: [email protected] (M. Dickmann), [email protected] (M. Marshall), [email protected] (F. Miraglia). 0168-0072/$ see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.apal.2004.06.005 28 M. Dickmann et al. / Annals of Pure and Applied Logic 132 (2005) 27–49