Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T07:25:53.808Z Has data issue: false hasContentIssue false
  • English
  • Français

The 116 reducts of (ℚ, <, a)

Published online by Cambridge University Press:  12 March 2014

Markus Junker  and
Martin Ziegler
Markus Junker
Affiliation:
Mathematisches Institut, Abteilung für Mathematische Logik, Universität Freiburg, Germany, E-mail: [email protected]
Martin Ziegler
Affiliation:
Mathematisches Institut, Abteilung für Mathematische Logik, Universität Freiburg, Germany, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This article aims to classify those reducts of expansions of (ℚ, <) by unary predicates which eliminate quantifiers, and in particular to show that, up to interdefinability, there are only finitely many for a given language. Equivalently, we wish to classify the closed subgroups of Sym(ℚ) containing the group of all automorphisms of (ℚ, <) fixing setwise certain subsets. This goal is achieved for expansions by convex predicates, yielding expansions by constants as a special case, and for the expansion by a dense, co-dense predicate. Partial results are obtained in the general setting of several dense predicates.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 861 - 884
Copyright
Copyright © Association for Symbolic Logic 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AZ] Ahlbrandt, Gisela and Ziegler, Martin, Invariant subgroups of v V, Journal of Algebra, vol. 151 (1992), no. 1, pp. 26–38.CrossRefGoogle Scholar
[B] Bennett, James, The redacts of some infinite homogeneous graphs and tournaments, Ph.D. thesis, Rutgers University, New Brunswick, 1996.Google Scholar
[C1] Cameron, Peter, Transitivity of permutation groups on unordered sets, Mathematische Zeitschrift, vol. 148 (1976), pp. 127–139.CrossRefGoogle Scholar
[C2] Cameron, Peter, Oligomorphic permutation groups, London Mathematical Society, Lecture Notes, vol.152, Cambridge University Press, Cambridge, 1990.CrossRefGoogle Scholar
[F] Frasnay, Claude, Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes, Annales de l'Institut Fourier (Grenoble), vol. 15 (1965), pp. 415–524.CrossRefGoogle Scholar
[Hi] Higman, Graham, Homogeneous relations, The Quarterly Journal of Mathematics, vol. 28 (1977), no. 109, pp. 31–39.CrossRefGoogle Scholar
[Ho] Hodges, Wilfrid, A shorter model theory, Cambridge University Press, Cambridge, 1997.Google Scholar
[HLS] Hodges, Wilfrid, Lachlan, Alistair, and Shelah, Saharon, Possible orderings of an indiscernible sequence, The Bulletin of the London Mathematical Society, vol. 9 (1977), pp. 212–215.CrossRefGoogle Scholar
[P1] Poizat, Bruno, Une théorie de Galois imaginaire, this Journal, vol. 48 (1983), no. 4, pp. 1151–1170.Google Scholar
[P2] Poizat, Bruno, A course in model theory, Springer, New York, 2000.CrossRefGoogle Scholar
[T1] Thomas, Simon, Reducts of the random graph, this Journal, vol. 56 (1991), pp. 176–181.Google Scholar
[T2] Thomas, Simon, Reducts of the random hypergraphs, Annals of Pure and Applied Logic, vol. 80 (1996), no. 2, pp. 165–193.CrossRefGoogle Scholar