Axiomatization of local-global principles for pp-formulas in spaces of orderings

Astier, Vincent; Tressl, Marcus

doi:10.1007/s00153-004-0236-0

Cited by 10 publications

(17 citation statements)

References 6 publications

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“…This is almost exactly the same result as the one proven by Astier and Tressl in [1], although their definition of the family K P slightly differs from ours. The implication 1.…”

supporting

confidence: 78%

“…Let (Y , H ) be the group extension of (Ỹ ,H) where H =H × {1, b 3 }. It consists of 6 orderings x We have thus finished a detailed description of K P and are in the position to build the formulae P One sees that P (1) 4 (y 1 , y 2 , y 3 ) contains 37 quantifiers and 40 atomic formulae. P (2) 4 (y 1 , y 2 , y 3 ) will contain, respectively, 473 quantifiers and 484 atomic formulae.…”

Section: For Everymentioning

confidence: 99%

“…⇒ 1. is more complicated, and we shall prove it in two different ways. The first proof is model-theoretic in nature and similar to the proof in [1], the second one will appear as a part of the proof of Theorem 2. Both proofs make use of the following lemma, utilized also in Astier and Tressl's original proof: Lemma 1.…”

mentioning

confidence: 99%

“…Trivially, for every space of orderings (X, G) and every a ∈ G k , P (a) ⇒ P (1) λ (a) (by taking z 0 = z 1 = . .…”

mentioning

confidence: 99%

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On families of testing formulae for a pp formula

Gładki¹,

Marshall²

2009

Quadratic Forms—Algebra, Arithmetic, and Geometry

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Astier and Tressl have recently proven that a pp formula fails on a finite subspace of a space of orderings if and only if a certain family of formulae is verified (V. Astier, M. Tressl, Axiomatization of localglobal principles for pp formulas in spaces of orderings, Arch. Math. Logic 44, No. 1 (2005), 77-95). The proof given in their paper is nonconstructive and uses rather advanced techniques from model theory. In this note we slightly strengthen their result by constructing another family of formulae with the same property, whose elements are given explicitly. We also illustrate the developed theory with an example of the testing family for a pp formula that is known to be a counterexample to the pp conjecture.

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“…This is almost exactly the same result as the one proven by Astier and Tressl in [1], although their definition of the family K P slightly differs from ours. The implication 1.…”

supporting

confidence: 78%

Section: For Everymentioning

confidence: 99%

mentioning

confidence: 99%

“…Trivially, for every space of orderings (X, G) and every a ∈ G k , P (a) ⇒ P (1) λ (a) (by taking z 0 = z 1 = . .…”

mentioning

confidence: 99%

See 2 more Smart Citations

On families of testing formulae for a pp formula

Gładki¹,

Marshall²

2009

Quadratic Forms—Algebra, Arithmetic, and Geometry

View full text Add to dashboard Cite

show abstract

“…They have already been studied, for example, in [Astier and Tressl 2005;Lira de Lima 1997;Mariano 2003], and notably in [Kula et al 1984], where the question of the realization of these special groups by fields is considered and where it is shown (as Corollary 4.7) that every profinite reduced special group is isomorphic to a quotient of the reduced special group of some field.…”

Section: Introductionmentioning

confidence: 99%