Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T03:57:31.993Z Has data issue: false hasContentIssue false
  • English
  • Français

REDUCTS OF STRUCTURES AND MAXIMAL-CLOSED PERMUTATION GROUPS

Published online by Cambridge University Press:  14 September 2016

MANUEL BODIRSKY  and
DUGALD MACPHERSON
MANUEL BODIRSKY
Affiliation:
INSTITUT FÜR ALGEBRA TU DRESDEN 01062 DRESDEN GERMANYE-mail:[email protected]
DUGALD MACPHERSON
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS LS2 9JT, UKE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Answering a question of Junker and Ziegler, we construct a countable first order structure which is not ω-categorical, but does not have any proper nontrivial reducts, in either of two senses (model-theoretic, and group-theoretic). We also construct a strongly minimal set which is not ω-categorical but has no proper nontrivial reducts in the model-theoretic sense.

Keywords

reductsmaximal closed subgroupsJordan permutation groupsD-relations
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 1087 - 1114
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

Adeleke, S. and Macpherson, H. D., Classification of infinite primitive Jordan permutation groups . Proceedings of the London Mathematical Society, vol. 72 (1996), no. 3, pp. 63–123.Google Scholar
Adeleke, S. and Neumann, P. M., Relations Related to Betweenness: Their Structure and Automorphisms, Memoirs of the American Mathematical Society, vol. 131, American Mathematical Society, Providence, RI, 1998.Google Scholar
Agarwal, L. and Kompatscher, M., 2ℵ0 pairwise non-isomorphic maximal-closed subgroups of Symℕ via the classificationw of the reducts of the Henson digraphs, arXiv:1509.07674.Google Scholar
Bennett, J. H., The reducts of some infinite homogeneous graphs and tournaments, Ph.D. thesis, Rutgers University, 1997.Google Scholar
Blossier, T., Automorphism groups of trivial strongly minimal structures, this Journal, vol. 68 (2003), pp. 644–668.Google Scholar
Bodirsky, M. and Pinsker, M., Reducts of Ramsey structures , Model Theoretic Methods in Finite Combinatorics, AMS Contemporary Mathematics, vol. 558, American Mathematical Society, Providence, RI, 2011, pp. 489–519.Google Scholar
Bogomolov, F. and Rovinsky, M., Collineation group as a subgroup of the symmetric group . European Journal of Mathematics, vol. 11 (2013) no. 1, pp. 17–26.Google Scholar
Cameron, P. J., Transitivity of permutation groups on unordered sets . Mathematische Zeitschrift, vol. 148 (1976), pp. 127–139.CrossRefGoogle Scholar
Cameron, P. J., Oligomorphic Permutation Groups, London Mathematical Society Lecture Note, vol. 152, Cambridge University Press, Cambridge, 1990.Google Scholar
Cherlin, G., Harrington, L., and Lachlan, A. H., ℵ0 -categorical, ℵ0 -stable structures. Annals of Pure and Applied Logic , vol. 28 (1985), pp. 103–135.Google Scholar
Chiswell, I., Introduction to Λ-Trees, World Scientific, Singapore, 2001.Google Scholar
Evans, D. M., Pillay, A., and Poizat, B., Le groupe dans le groupe . Algebra i Logika, vol. 29 (1990) no. 3, pp. 368–378. Translation in Algebra and Logic , vol. 29(1990), no. 3, pp. 244–252.Google Scholar
Hodges, W. A., Model Theory, Cambridge University Press, Cambridge, 1993.Google Scholar
Hrushovski, E., Locally modular regular types , Classification Theory (Proceedings, Chicago, 1985), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1987, pp. 132–164.Google Scholar
Junker, M. and Ziegler, M., The 116 reducts of (ℚ,<,a), this Journal, vol. 73 (2008), pp. 861–884.Google Scholar
Kaplan, I. and Simon, P., The affine and projective groups are maximal . Transactions of the American Mathematical Society, to appear.Google Scholar
Macpherson, H. D., Infinite distance transitive graphs of finite valency . Combinatorica, vol. 2 (1982), pp. 63–69.Google Scholar
Macpherson, H. D., A survey of Jordan groups , Automorphisms of First Order Structures (Kaye, R. and Macpherson, H. D., editors), Clarendon Press, Oxford, 1994.Google Scholar
Macpherson, H. D. and Neumann, P. M., Subgroups of infinite symmetric groups . Journal of the London Mathematical Society, vol. 42 (1990), no. 2, pp. 64–84.Google Scholar
Macpherson, H. D. and Steinhorn, C., On variants of o-minimality . Annals of Pure and Applied Logic, vol. 79 (1996), pp. 165–209.Google Scholar
Marker, D. and Pillay, A., Reducts of 〈ℂ,+, .〉 that contain +, this Journal, vol. 55 (1990), pp. 1243–1255.Google Scholar
Mortimer, B., Permutation groups containing affine groups of the same degree . Journal of the London Mathematical Society, vol. 12 (1980), pp. 303–307.Google Scholar
Nešetřil, J., Ramsey classes and homogeneous structures . Combinatorics, Probability and Computing, vol. 14 (2005), pp. 171–189.Google Scholar
Neumann, P. M., Some primitive permutation groups . Proceedings of the London Mathematical Society, vol. 50 (1985), no. 3, pp. 265–281.Google Scholar
Pach, P. P., Pinsker, M., Pluhár, G., Pongrácz, A., and Szabó, Cs., Reducts of the random partial order . Advances in Mathematics, vol. 267 (2014), pp. 94–120.Google Scholar
Parigot, M., Théories d’arbres, this Journal, vol. 47 (1982), pp. 841–853.Google Scholar
Peterzil, Y., Reducts of some structures over the reals, this Journal, vol. 58 (1993), pp. 955–966.Google Scholar
Pillay, A., Geometric stability theory , Oxford Logic Guides, vol. 32, Clarendon Press, Oxford, 1996.Google Scholar
Shelah, S., Classification Theory and the Number of Non-isomorphic Models, second ed., North-Holland, Amsterdam, 1990.Google Scholar
Tent, K. and Ziegler, M., A Course in Model Theory, Lecture Notes in Logic, vol. 40, Cambridge University Press, Cambridge, 2012.Google Scholar
Thomas, S., Reducts of the random graph, this Journal, vol. 56 (1991), pp. 176–181.Google Scholar
Thomas, S., Reducts of random hypergraphs . Annals of Pure and Applied Logic, vol. 80 (1996), pp. 165–193.Google Scholar
Zilber, B. I., Uncountably Categorical Theories, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1993.Google Scholar