Direct limits of finite spaces of orderings

Kula, Mieczysław; Marshall, Murray; Sładek, Andrzej

doi:10.2140/pjm.1984.112.391

Cited by 9 publications

(4 citation statements)

References 4 publications

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“…Every profinite RSG is lattice ordered.Proof. In his (not yet published) thesis[13] (Theorem 2.50, p. 74) Mariano proves, generalizing a previous result from[9] (Lemma 4.4), that any profinite structure (in any language L) is a retract of an ultraproduct of finite structures. To be precise, given a projective system (M i ; f j i ) : i ≤ j ; i, j ∈ I of finite L-structures over a right-directed set I, ≤ , with projective limit M, and an upper-directed ultrafilter D (that is, an ultrafilter containing the sets…”

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confidence: 58%

Lattice-ordered reduced special groups

Dickmann¹,

Marshall²,

Miraglia³

2005

Annals of Pure and Applied Logic

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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.

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confidence: 58%

Lattice-ordered reduced special groups

Dickmann¹,

Marshall²,

Miraglia³

2005

Annals of Pure and Applied Logic

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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].…”

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confidence: 87%

“…The quotient structure (X 0 , G 0 ) with G 0 = ker γ is not a quotient space. (4) Consider the space (X, G), where 8 . This is the direct sum of two connected spaces, each consisting of six elements.…”

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confidence: 99%

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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“…Profinite special groups where introduced in [22], generalizing results in [23], obtained in the setting of abstract order spaces, a category that is dual to that of reduced special groups (cf. Theorem 3.19, p. 57, in [9]).…”

Section: The Profinite Hull Of a Reduced Special Groupmentioning

confidence: 99%

The Profinite Hull of Special Groups and Local-Global Principles

Mariano

Miraglia

2010

Stud Logica

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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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Direct limits of finite spaces of orderings

Cited by 9 publications

References 4 publications

Lattice-ordered reduced special groups

Lattice-ordered reduced special groups

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

The Profinite Hull of Special Groups and Local-Global Principles

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