Abstract
We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Edmundo M.: Solvable groups definable in o-minimal structures. J. Pure Appl. Algebra 185(1–3), 103–145 (2003)
Edmundo M.: Covers of groups definable o-minimal structures. Ill. J. Math. 49(1), 99–120 (2005)
Edmundo M.: A remark on divisibility of definable groups. Math. Logic. Quart. 51(6), 639–641 (2005)
Edmundo M.: Locally definable groups in o-minimal structures. J. Algebra 301(1), 194–223 (2006)
Edmundo M.: Covers of groups definable o-minimal structures. Errat. Ill. J. Math. 51(3), 1037–1038 (2007)
Edmundo M.: Locally definable groups in o-minimal structures. Corrigendum (with E. Baro) J. Algebra 320(7), 3079–3080 (2008)
Edmundo M., Eleftheriou P.: The universal covering homomorphism in o-minimal expansions of groups. Math. Log. Quart. 53(6), 571–582 (2007)
Edmundo M., Otero M.: Definably compact abelian groups. J. Math. Logic 4(2), 163–180 (2004)
Edmundo M., Jones G., Peatfield N.: Sheaf cohomology in o-minimal structures. J. Math. Logic 6(2), 163–179 (2006)
Eleftheriou, P.: Groups definable in linear o-minimal structures. Ph.D. Thesis, University of Notre Dame (2007)
Eleftheriou P., Starchenko S.: Groups definable in ordered vector spaces over ordered division rings. J. Symb. Logic 72, 1108–1140 (2007)
Fulton W.: Algebraic Topology. Springer, Berlin (1995)
Hrushovski, E., Pillay, A.: On NIP and invariant measures. Preprint Jan (2009)
Hrushovski E., Peterzil Y., Pillay A.: Groups, measures and the NIP. J. Amer. Math. Soc. 21(2), 563–596 (2008)
Hrushovski, E., Peterzil, Y., Pillay, A.: On central extensions and definably compact groups in o-minimal structures. Preprint Oct (2008)
Otero, M., Peterzil, Y.: G-linear sets and torsion points in definably compact groups. Archive Math. Logic (to appear)
Peterzil, Y.: Returning to semi-bounded sets. Preprint Oct (2007)
Peterzil Y., Starchenko S.: Definable homomorphisms of abelian groups definable in o-minimal structures. Ann. Pure Appl. Logic 101(1), 1–27 (1999)
Peterzil Y., Pillay A., Starchenko S.: Definably simple groups in o-minimal structures. Trans. Amer. Math. Soc. 352(10), 4397–4419 (2000)
Peterzil Y., Pillay A., Starchenko S.: Simple algebraic and semialgebraic groups over real closed fields. Trans. Amer. Math. Soc. 352(10), 4421–4450 (2000)
Peterzil Y., Pillay A., Starchenko S.: Linear groups definable in o-minimal structures. J. Algebra 247, 1–23 (2002)
Pillay A.: On groups and fields definable in o-minimal structures. J. Pure Appl. Algebra 53, 239–255 (1988)
Pillay A.: Type-definability, compact Lie groups and o-minimality. J. Math. Logic 4(2), 147–162 (2004)
van den Dries L.: Tame Topology and O-minimal Structures. Cambridge University Press, Cambridge, MA (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mário J. Edmundo—With partial support from the FCT (Fundação para a Ciência e Tecnologia) program POCI 2010 (Portugal/FEDER-EU) and FCT (Fundação para a Ciência e Tecnologia) project PTDC/MAT/101740/2008.
Gareth O. Jones—Supported by NSERC while a post-doc at McMaster University, Canada.
Rights and permissions
About this article
Cite this article
Edmundo, M.J., Jones, G.O. & Peatfield, N.J. Invariance results for definable extensions of groups. Arch. Math. Logic 50, 19–31 (2011). https://doi.org/10.1007/s00153-010-0196-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-010-0196-5