Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-27T13:36:56.290Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the model theory of generalized polygons

Published online by Cambridge University Press:  12 March 2014

Katrin Tent
Katrin Tent*
Affiliation:
Mathematisches Institut, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Using projectivity groups, we classify some polygons with strongly minimal point rows and show in particular that no infinite quadrangle can have sharply 2-transitive projectivity groups in which the point stabilizers are abelian. In fact, we characterize the finite orthogonal quadrangles Q(4,2). Q− (5.2) and Q(4,3) by this property. Finally we show that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 2 , June 2000 , pp. 692 - 702
Copyright
Copyright © Association for Symbolic Logic 2000

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[B1]Baldwin, J. T., An almost strongly minimal non-desarguesian projective plane, Transactions of the American Mathematical Society, vol. 342 (1994), pp. 695–711.CrossRefGoogle Scholar
[B2]Baldwin, J. T., Some projective planes of Lenz-Barlotti class I, Proceedings of the American Mathematical Society, vol. 123 (1995).CrossRefGoogle Scholar
[BH]Baldwin, J. T. and Holland, K., Constructing ω-stable structures: Rank 2 fields, preprint.Google Scholar
[BK]Bödi, R. and Kramer, L., On homomorphisms between generalized polygons, Geometriae Dedicata, vol. 58 (1995), pp. 1–14.CrossRefGoogle Scholar
[BN]Borovik, A. and Nesin, A., Groups of finite Morley rank, Oxford science publication, 1994.CrossRefGoogle Scholar
[BTVM]Brouns, L., Tent, K., and van Maldeghem, H., Groups of projectivities of generalized quadrangles, Geometriae Dedicata, to appear.Google Scholar
[Dem]Dembowski, P., Finite geometries, Springer, 1968.CrossRefGoogle Scholar
[HM]Hanssens, G. and van Maldeghem, H., Algebraic properties of quadratic quaternary rings, Geometriae Dedicata, vol. 30 (1989), pp. 43–67.CrossRefGoogle Scholar
[Hr]Hrushovski, E., Almost orthogonal regular types, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 139–155.CrossRefGoogle Scholar
[Ker]Kerby, W., On infinite sharply multiply transitive groups, Hamburg Mathematische Einzelschriften, Neue Folge, Heft 6, Göttingen, 1968.Google Scholar
[Kn1]Knarr, N., Projectivities of generalized polygons, Ars Combinatoria, vol. 25B (1988), pp. 265–275.Google Scholar
[Kn2]Knarr, N., The nonexistence of certain topological polygons, Forum Mathematicum, vol. 2 (1990), pp. 603–612.CrossRefGoogle Scholar
[KT]Kramer, L. and Tent, K., Algebraic polygons, Journal of Algebra, vol. 182 (1996), pp. 435–447.CrossRefGoogle Scholar
[KTVM]Kramer, L., Tent, K., and van Maldeghem, H., Simple groups of finite Morley rank and Tits buildings, Israel Journal of Math, vol. 109 (1999), pp. 189–224.CrossRefGoogle Scholar
[VM]van Maldeghem, H., Generalized polygons, a geometric approach, Birkhäuser, 1998.Google Scholar
[N]Nesin, A., On bad-groups, bad-fields and pseudo-planes, this Journal, vol. 56 (1991), pp. 915–931.Google Scholar
[Pi]Pickert, G., Projectivities in projective planes, Geometry—von Staudt's point of view (Plaumann, P. and Strambach, K., editors), Reidel, 1981, pp. 1–49.Google Scholar
[Te]Tent, K., Very homogeneous generalized n-gons of finite Morley Rank, Journal of the London Mathematical Society, to appear.Google Scholar
[Th]Thas, J. A., Generalized Polygons, Handbook of incidence geometry (Buekenhout, F., editor), North-Holland, 1995, pp. 383–431.CrossRefGoogle Scholar
[W]Weiss, R., The nonexistence of certain Moufang polygons, Inventiones Mathematicae, vol. 51 (1979), pp. 261–266.CrossRefGoogle Scholar