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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]
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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

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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