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Generic automorphisms of the universal partial order

Authors: D. Kuske and J. K. Truss
Journal: Proc. Amer. Math. Soc. 129 (2001), 1939-1948
MSC (2000): Primary 06A06, 20B27
Published electronically: December 13, 2000
MathSciNet review: 1825900
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Abstract: We show that the countable universal-homogeneous partial order $(P,<)$ has a generic automorphism as defined by the second author, namely that it lies in a comeagre conjugacy class of Aut$(P,<)$. For this purpose, we work with `determined' partial finite automorphisms that need not be automorphisms of finite substructures (as in the proofs of similar results for other countable homogeneous structures) but are nevertheless sufficient to characterize the isomorphism type of the union of their orbits.

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

  • 1. A. M. W. Glass, S. H. McCleary and M. Rubin, Automorphism groups of countable highly homogeneous partially ordered sets. Math. Zeit. 214 (1993), 55-66. MR 94i:20005
  • 2. Bernhard Herwig, Extending partial isomorphisms on finite structures, Combinatorica 15 (1995), 365-371. MR 97a:03044
  • 3. Bernhard Herwig, Extending partial isomorphisms for the small index property of many $\omega$-categorical structures, Israel Journal of Mathematics, 107 (1998), 93-123. MR 2000c:03027
  • 4. Bernhard Herwig and Daniel Lascar, Extending partial automorphisms and the profinite topology on free groups, to appear in the Transactions of the Amer. Math. Soc. CMP 98:12
  • 5. Wilfrid Hodges, Ian Hodkinson, Daniel Lascar, and Saharon Shelah, The small index property for $\omega$-categorical $\omega$-stable structures and for the random graph, Journal of the London Math. Soc. (2) 48 (1993), 204-218. MR 94d:03063
  • 6. I. M. Hodkinson, personal communication.
  • 7. Ehud Hrushovski, Extending partial isomorphisms of graphs, Combinatorica 12 (1992), 204-218. MR 93m:05089
  • 8. M. Rubin, The reconstruction of boolean algebras from their automorphism groups, in Handbook of Boolean algebras vol. 2, (edited by J.D.Monk, R.Bonnet), Elsevier (1989), 549-606. CMP 21:10
  • 9. M. Rubin, The reconstruction of trees from their automorphism groups, Contemporary Mathematics 151 (1993), Amer. Math. Soc. MR 95f:20046
  • 10. J. Schmerl, Countable homogeneous partially ordered sets, Algebra Universalis 9, (1979), 317-321. MR 81g:06001
  • 11. J.K.Truss, Generic automorphisms of homogeneous structures, Proceedings of the London Math. Soc. 64 (1992), 121-141. MR 93f:20008

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Additional Information

D. Kuske
Affiliation: Institut für Algebra, Technische Universität Dresden, D-01062 Dresden, Germany
Address at time of publication: Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, England

J. K. Truss
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England

Keywords: Generic, partially ordered set, universal homogeneous, partial automorphism
Received by editor(s): September 14, 1999
Received by editor(s) in revised form: November 15, 1999
Published electronically: December 13, 2000
Additional Notes: Research supported by a grant from the British-German Academic Collaboration Programme. The authors thank the referee for helpful suggestions.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2000 American Mathematical Society

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