Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-12T22:28:35.777Z Has data issue: false hasContentIssue false
  • English
  • Français

Recovering ordered structures from quotients of their automorphism groups

Published online by Cambridge University Press:  12 March 2014

M. Giraudet  and
J. K. Truss
M. Giraudet
Affiliation:
Université du Maine, Faculté des Sciences, Département de Mathématiques, Avenue Olivier Messiaen, 72085 le Mans Cedex 9, France, E-mail: [email protected]
J. K. Truss
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the ‘tail’ of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

Keywords

chaindoubly homogeneous chaincountable cofinalityquotientrecovery
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1189 - 1198
Copyright
Copyright © Association for Symbolic Logic 2003

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

[1] Droste, Manfred, Giraudet, Michèle, and Göbel, Rüdiger, All groups are outer automorphism groups of simple groups, Journal of the London Mathematical Society, vol. 64 (2001), pp. 565–575.Google Scholar
[2] Droste, Manfred and Shelah, Saharon, Outer automorphism groups of ordered permutation groups, Forum Mathematicum, vol. 14 (2002), pp. 605–621.CrossRefGoogle Scholar
[3] Giraudet, M. and Truss, J. K., On distinguishing quotients of ordered permutation groups, The Quarterly Journal of Mathematics, vol. 45 (1994), pp. 181–209.CrossRefGoogle Scholar
[4] Glass, A. M. W., Ordered permutation groups, London Mathematical Society Lecture Note Series, vol. 55, Cambridge University Press, 1981.Google Scholar
[5] Glass, A. M. W., Gurevich, Yuri, Holland, W. Charles, and Jambu-Giraudet, Michèle, Elementary theory of automorphism groups of doubly homogeneous chains, Logic year 1979–1980, Lecture Notes in Mathematics, vol. 859, Springer, 1981, pp. 67–82.Google Scholar
[6] Jambu-Giraudet, Michèle, Théorie des modèles de groupes d'automorphismes d'ensembles totalement ordonnés, Universite de Paris VII, 1979, Thèse de troisième cycle.Google Scholar
[7] Jambu-Giraudet, Michèle, Interprétations d'arithmétiques dans des groupes et des treillis, Model theory and arithmetic proceedings, Lecture Notes in Mathematics, vol. 890, Springer, 1981, pp. 143–153.Google Scholar
[8] Jambu-Giraudet, Michèle, Bi-interpretable groups and lattices, Transactions of the American Mathematical Society, vol. 278 (1983), pp. 253–269.Google Scholar
[9] Jambu-Giraudet, Michèle, Quelques remarques sur l'équivalence élémentaire entre groupes ou treillis d'automorphismes de chaînes 2-homogènes, Discrete Mathematics, vol. 53 (1985), pp. 117–124.Google Scholar
[10] Truss, J. K., Infinite permutation groups, Products ofconjugacy classes, Journal of Algebra, vol. 120 (1989), pp. 454–493.Google Scholar