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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]
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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

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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