Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T08:18:28.435Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE

Part of: Connections with other structures, applications Permutation groups Ordered sets Structure and classification of infinite or finite groups Set theory Model theory

Published online by Cambridge University Press:  01 February 2021

ITAY KAPLAN ,
TOMASZ RZEPECKI  and
DAOUD SINIORA
ITAY KAPLAN
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM 91904, JERUSALEM, ISRAELE-mail: kaplan@math.huji.ac.il
TOMASZ RZEPECKI
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM 91904, JERUSALEM, ISRAEL and INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4 50-384 WROCŁAW, POLANDE-mail: tomasz.rzepecki@math.uni.wroc.pl
DAOUD SINIORA
Affiliation:
INDEPENDENT SCHOLAR E-mail: daoud.siniora@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

Keywords

Fraïssé limitgeneric automorphismtreePolish groupample generics

MSC classification

Primary: 03C15: Denumerable structures
Secondary: 03E15: Descriptive set theory 06A12: Semilattices 20E08: Groups acting on trees 54H11: Topological groups 20B27: Infinite automorphism groups
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1508 - 1540
Check for updates
Copyright
© Association for Symbolic Logic 2021

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

Bodirsky, M., Bradley-Williams, D., Pinsker, M., and Pongrácz, A., The universal homogeneous binary tree . Journal of Logic and Computation, vol. 28 (2018), no. 1, pp. 133–163.10.1093/logcom/exx043CrossRefGoogle Scholar
Droste, M., Holland, W. C., and Macpherson, H. D., Automorphism groups of infinite semilinear orders (II) . Proceedings of the London Mathematical Society, vol. s3-58 (1989), no. 3, pp. 479–494.10.1112/plms/s3-58.3.479CrossRefGoogle Scholar
Duchesne, B., Topological properties of Ważewski dendrite groups . Journal de l’École polytechnique—Mathématiques, vol. 7 (2020), pp. 431–477.10.5802/jep.121CrossRefGoogle Scholar
Hodges, W., A Shorter Model Theory, Cambridge University Press, New York, NY, 1997.Google Scholar
Ivanov, A. A., Generic expansions of $\omega$ -categorical structures and semantics of generalized quantifiers. this Journal, vol. 64 (1999), no. 2, pp. 775–789.Google Scholar
Kaplan, I. and Shelah, S., A dependent theory with few indiscernibles . Israel Journal of Mathematics, vol. 202 (2014), no. 1, pp. 59–103.10.1007/s11856-014-1067-2CrossRefGoogle Scholar
Kaplan, I. and Shelah, S., Examples in dependent theories , this Journal, vol. 79 (2014), no. 2, pp.585–619.Google Scholar
Kechris, A. S., Pestov, V. G., and Todorcevic, S., Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geometric & Functional Analysis, vol. 15 (2005), no. 1, pp. 106–189.10.1007/s00039-005-0503-1CrossRefGoogle Scholar
Kechris, A. S. and Rosendal, C., Turbulence, amalgamation, and generic automorphisms of homogeneous structures . Proceedings of the London Mathematical Society, vol. 94 (2007), no. 2, pp. 302–350.10.1112/plms/pdl007CrossRefGoogle Scholar
Kuske, D. and Truss, J. K., Generic automorphisms of the universal partial order . Proceedings of the American Mathematical Society, vol. 129 (2001), no. 7, pp. 1939–1948.CrossRefGoogle Scholar
Kwiatkowska, A., The group of homeomorphisms of the cantor set has ample generics . Bulletin of the London Mathematical Society, vol. 44 (2012), no. 6, pp. 1132–1146.10.1112/blms/bds039CrossRefGoogle Scholar
Kwiatkowska, A. and Malicki, M., Ordered structures and large conjugacy classes . Journal of Algebra, vol. 557 (2020), pp. 67–96.10.1016/j.jalgebra.2020.03.021CrossRefGoogle Scholar
Macpherson, D., A survey of homogeneous structures . Discrete Mathematics, vol. 311 (2011), no. 15, pp. 1599–1634.10.1016/j.disc.2011.01.024CrossRefGoogle Scholar
Simon, P., A Guide to NIP Theories , Lecture Notes in Logic, vol. 44, Association for Symbolic Logic and Cambridge Scientific, Chicago, IL and Cambridge, 2015.Google Scholar
Simon, P., NIP omega-categorical structures: The rank 1 case, arXiv preprint, 2018, arXiv:1807.07102.Google Scholar
Siniora, D. N., Automorphism groups of homogeneous structures , Ph.D. thesis, University of Leeds, 2017.Google Scholar
Truss, J. K., Generic automorphisms of homogeneous structures . Proceedings of the American Mathematical Society, vol. s3-65 (1992), no. 1, pp. 121–141.10.1112/plms/s3-65.1.121CrossRefGoogle Scholar
Truss, J. K., On notions of genericity and mutual genericity , this Journal, vol. 72 (2007), no. 3, pp. 755–766.Google Scholar