Elementary properties of the Boolean hull and reduced quotient functors

In [12] we proved the following isotropy-reflection principle: Theorem. Let F be a formally real field and let F p denote its Pythagorean closure. The natural embedding of reduced special groups from G red ( F ) into G red (F p ) = G(F P ) induced by the inclusion of fields, reflects isotropy . Here G red ( F ) denotes the reduced special group (with underlying group Ḟ/ΣḞ 2 ) associated to the field F , henceforth assumed formally real; cf. [11], Chapter 1, §3, for details. The result proved in [12] is, in fact, more general. For example, the Pythagorean closure F p can be replaced in the statement above by the intersection of all real closures of F (inside a fixed algebraic closure). Similar statements hold, more generally, for all relative Pythagorean closures of F in the sense of Becker [3], Chapter II, §3. Since the notion of isotropy of a quadratic form can be expressed by a first-order formula in the natural language L SG for special groups (with the coefficients as parameters), this result raises the question whether the embedding ι FFp : G red ( F ) ↪ G ( F p ) is elementary. Further, since the L SG -formula expressing isotropy is positive-existential, one may also ask whether ι FF p reflects all (closed) formulas ofthat kind with parameters in G red ( F ). In this paper we give a negative answer to the first of these questions, for a vast class of formally real (non-Pythagorean) fields F (Prop. 5.1). This follows from rather general preservation results concerning the “Boolean hull” and the “reduced quotient” operations on special groups.