Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$\mathbb {Q}$ muni de l'arithmétique faible
de Penzin est décidable

Author: Françoise Delon
Journal: Proc. Amer. Math. Soc. 125 (1997), 2711-2717
MSC (1991): Primary 03C60; Secondary 12L05
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the decidability of the additive ordered group $\mathbb {Q}$ equipped with a predicate for $2^{\mathbb {Z}}$, the multiplication restricted to $2^{\mathbb {Z}}\times \mathbb {Q}$ and the $2$-adic valuation ranging in $2^{\mathbb {Z}}$.

References [Enhancements On Off] (What's this?)

  • [BHMV] Véronique Bruyère, Georges Hansel, Christian Michaux, and Roger Villemaire, Logic and 𝑝-recognizable sets of integers, Bull. Belg. Math. Soc. Simon Stevin 1 (1994), no. 2, 191–238. Journées Montoises (Mons, 1992). MR 1318968
    Véronique Bruyère, Georges Hansel, Christian Michaux, and Roger Villemaire, Correction to: “Logic and 𝑝-recognizable sets of integers”, Bull. Belg. Math. Soc. Simon Stevin 1 (1994), no. 4, 577. MR 1315840
  • [CP] B. I. Dahn (ed.), Proceedings of the fourth Easter conference on model theory, Seminarberichte [Seminar Reports], vol. 86, Humboldt Universität, Sektion Mathematik, Berlin, 1986. Held in Gross Köris, March 31–April 5, 1986. MR 884714
  • [DS] F. Delon et P. Simonetta, Un principe d'Ax-Kochen-Ershov pour des structures intermédiaires entre groupes et corps valués, JSL, à paraître.
  • [G] Joachim Gräter, Lokalinvariante Bewertungen, Math. Z. 192 (1986), no. 2, 183–194 (German). MR 840822,
  • [GH] C. Grünenwald et F. Haug, On stable groups in some soluble group classes, Seminarbericht 93-1 der Humboldt-Universität zu Berlin (Mathematik), Berlin, 1993, pp. 46-59.
  • [P] Y. Penzin, Solvability of the theory of integers with addition, order, and multiplication by an arbitrary number, Math Notes Academy of Sciences USSR (1973), 401-405.
  • [Ra] A. Robinson, Solution of a problem of Tarski, Fund. Math. 47 (1959), 179–204. MR 0112841
  • [Rj] Julia Robinson, Definability and decision problems in arithmetic, JSL 14 (1949), 98-114. MR 11:151f
  • [Rr] Raphael Robinson, Undecidable rings, TAMS 70 (1951), 137-159. MR 12:791b
  • [S1] P. Simonetta, Une correspondance entre anneaux partiels et groupes, J.S.L., à paraître.
  • [S2] P. Simonetta, Décidabilité et équivalence élémentaire pour des structures du type groupe agissant sur un groupe, JSL, à paraître.
  • [vdD1] L. van den Dries, Model Theory of Fields, Thèse, Utrecht, 1978.
  • [vdD2] Lou van den Dries, The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985), no. 1-2, 187–195. MR 808687,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03C60, 12L05

Retrieve articles in all journals with MSC (1991): 03C60, 12L05

Additional Information

Françoise Delon
Affiliation: CNRS-Université Paris 7, UFR de Mathématiques, 2 place Jussieu, 75251 Paris cedex 05, France

Keywords: Penzin arithmetic, decidability, field of rational numbers, $p$-adic valuation
Received by editor(s): September 20, 1995
Received by editor(s) in revised form: April 4, 1996
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1997 American Mathematical Society