Algebra universalis

, 79:92 | Cite as

Fractional parts of dense additive subgroups of real numbers

  • Luc Bélair
  • Françoise PointEmail author


Given a dense additive subgroup G of \(\mathbb {R}\) containing \(\mathbb {Z}\), we consider its intersection \(\mathbb {G}\) with the interval [0, 1[ with the induced order and the group structure given by addition modulo 1. We axiomatize the theory of \(\mathbb {G}\) and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a standard part and two ordered semigroups of infinitely small and infinitely large elements.


Dense subgroups Ordering Model-completeness 

Mathematics Subject Classification

00A99 08A40 06E30 



  1. 1.
    Bélair, L., Point, F.: La logique des parties fractionnaires de nombres réels. C. R. Acad. Sci. Paris Ser. I(354), 645–648 (2016)CrossRefGoogle Scholar
  2. 2.
    Boigelot, B., Rassart, S., Wolper, P.: On the expressiveness of real and integer arithmetic automata. In: Proceeding ICALP ’98, Proceedings of the 25th International Colloquium on Automata, Languages and Programming. Lect. Notes in Comp. Sci., vol. 1443, pp. 152–163. Springer (1998)Google Scholar
  3. 3.
    Bouchy, F., Finkel, A., Leroux, J.: Decomposition of decidable first-order logics over integers and reals. In: Temporal Representation and reasoning, proceedings of the 15th symposium TIME 2008, pp. 147–155. IEEE Computer Society Press (2008)Google Scholar
  4. 4.
    Clifford, A.: Totally ordered commutative semigroups. Bull. Am. Math. Soc. 64, 305–316 (1958)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dolich, A., Goodrick, J.: Strong theories of ordered abelian groups. Fundam. Math. 236(3), 269–296 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Giraudet, M., Leloup, G., Lucas, F.: First-order theory of cyclically ordered groups. Ann. Pure Appl. Logic 169(12), 896–927 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Günaydin, A.: Model theory of fields with multiplicative groups. Ph.D. thesis, University of Illinois at Urbana-Champaign (2008)Google Scholar
  8. 8.
    Gurevich, Y., Schmitt, P.H.: The theory of ordered abelian groups does not have the independence property. Trans. Am. Math. Soc. 284(1), 171–182 (1984)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hall, M.: The Theory of Groups. Macmillan, New York (1959)zbMATHGoogle Scholar
  10. 10.
    Ibuka, S., Kikyo, H., Tanaka, H.: Quantifier elimination for lexicographic products of ordered abelian groups. Tsukuba J. Math. 33, 95–129 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaplan, I., Shelah, S.: Chain conditions in dependent groups. Ann. Pure Appl. Logic 164(12), 1322–1337 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leloup, G.: Autour des groupes cycliquement ordonnés. Ann. Fac. Sci. Toulouse Math. (6) 21(2), 235–257 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Leloup, G., Lucas, F.: c-regular cyclically ordered groups. arXiv:1312.5269 (2013)
  14. 14.
    Lucas, F.: Théorie des modèles des groupes cycliquement ordonnés divisibles. In: Séminaire de structures algébriques ordonnées, Prépublications de l’Université Paris VII 56 (1996)Google Scholar
  15. 15.
    Lucas, F.: Théorie des modèles de groupes abéliens ordonnés. Thèse d’habilitation à diriger des recherches, Université Paris VII (1996)Google Scholar
  16. 16.
    Marker, D.: Model Theory: An Introduction. Springer, New York (2002)zbMATHGoogle Scholar
  17. 17.
    Miller, C.: Expansions of dense linear orders with the intermediate value proper. J. Symb. Logic 66, 1783–1790 (2001)CrossRefGoogle Scholar
  18. 18.
    Robinson, A., Zakon, E.: Elementary properties of ordered abelian groups. Trans. Am. Math. Soc. 96, 222–236 (1960)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shelah, S.: Minimal bounded index subgroup for dependent theories. Proc. Am. Math. Soc. 136(3), 1087–1091 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Świerczkowski, S.: On cyclically ordered groups. Fundam. Math. 47, 161–166 (1959)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Weispfenning, V.: Mixed real-integer linear quantifier elimination. In: Symbolicand algebraic computation, proceedings ISSAC’99, pp. 129–136. ACM, Vancouver (1999)Google Scholar
  22. 22.
    Weispfenning, V.: Elimination of quantifiers for certain ordered and lattice-ordered abelian groups. In: Proceedings of the Model Theory Meeting (Univ. Brussels, Brussels/Univ. Mons, Mons, 1980). Bull. Soc. Math. Belg. Sér. B, vol. 33 no. 1, pp. 131–155 (1981)Google Scholar
  23. 23.
    Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26, 149–213 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité du Québec-UQAMMontrealCanada
  2. 2.Département de MathématiqueUniversité de Mons, De VinciMonsBelgium

Personalised recommendations