Realizing profinite reduced special groups

We introduce the Profinite Hull functor of special groups, showing that it gives rise to a new (and strong) local-global principle, the subform reflection property. We also indicate applications of this principle to the abstract algebraic theory of quadratic forms.The theory of special groups (SGs) is a first-order axiomatization of the algebraic theory of quadratic forms, introduced by M. Dickmann and F. Miraglia (see [9]). The local-global principles in the algebraic theory of quadratic forms were developed, initially, in the context of fields. Of fundamental importance is Pfister's local-global principle for isometry in the reduced theory of quadratic forms, a vast generalization of Sylvester's inertia law for forms with unit real coefficients. This was in turn generalized to reduced special groups context in [9]. There are other interesting abstract formulations of the reduced theory of quadratic forms, as the theory of abstract order spaces introduced M. Marshall (see [26]), that is shown in [9] to be essentially equivalent to the theory of reduced special groups (RSGs).The Boolean hull of a RSG is introduced and studied in [9], being an essential ingredient in the solution of Marshall's signature conjecture in [10]. The profinite hull functor introduced here, defined on the category of RSGs, is, in some sense, a refinement of the Boolean hull construction, yielding a new (and stronger) local-global principle, the subform reflection property.The paper is divided into three sections. In section 1 we recall basic results on special groups, needed in the sequel. We also prove two new results: the first gives a very useful sufficient condition for the reflection of Special Issue: The Legacy of Newton da Costa Edited by Daniele Mundici and Itala M. Loffredo D'Ottaviano

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“…. , β n ≡ G v ⊕ χ, that, together with the first isometry in (5), completes the proof of (b) and of Fact 6.…”

Section: Monomorphisms In the Category Of Special Groupssupporting

confidence: 58%

“…In the forthcoming paper [5] it is shown that any profinite reduced special group is isomorphic to the special group associated to a Pythagorean field.…”

Section: Profinite Special Groupsmentioning

confidence: 99%

The Profinite Hull of Special Groups and Local-Global Principles

Mariano

Miraglia

2010

Stud Logica

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“…At the same time, it remains an open problem whether profinite spaces of orderings are realizable. We note here that the dual question of whether direct limits of finite spaces of orderings are realizable was partially answered already in the early 1980s in [8], and recently completely resolved in [1].…”

mentioning

confidence: 88%

Quotients of index two and general quotients in a space of orderings

Gładki

Marshall

2015

Fund. Math.

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In our work we investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field Q(x) is particularly interesting, since then the theory developed simplifies significantly. A part of the theory firstly developed for quotients of index 2 generalizes in an elegant way to quotients of index 2^n for arbitrary finite n. Numerous examples are provided

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The Boolean and profinite hulls of reduced special groups

Mariano¹,

Miraglia²

2012

Logic Journal of IGPL

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Realizing profinite reduced special groups

Cited by 3 publications

References 13 publications

The Profinite Hull of Special Groups and Local-Global Principles

The Profinite Hull of Special Groups and Local-Global Principles

Quotients of index two and general quotients in a space of orderings

The Boolean and profinite hulls of reduced special groups

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