scholarly journals MOUFANG QUADRANGLES OF MIXED TYPE

2008 ◽  
Vol 50 (1) ◽  
pp. 143-161
Author(s):  
KOEN STRUYVE ◽  
HENDRIK VAN MALDEGHEM
Keyword(s):  
Mixed Type ◽  
Geometric Characterization ◽  
Weak Version ◽  
Generalized Quadrangles

AbstractIn this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author.

scholarly journals The inner and outer space of 2-dimensional Laguerre planes

1997 ◽  
Vol 62 (1) ◽  
pp. 104-127 ◽  
Author(s):  
B. Polster ◽  
G. F. Steinke
Keyword(s):  
Outer Space ◽  
Laguerre Plane ◽  
Generalized Quadrangles ◽  
Base Points ◽  
Laguerre Planes

AbstractThe classical 2-dimensional Laguerre plane is obtained as the geometry of non-trivial plane sections of a cylinder in R3 with a circle in R2 as base. Points and lines in R3 define subsets of the circle set of this geometry via the affine non-vertical planes that contain them. Furthemore, vertical lines and planes define partitions of the circle set via the points and affine non-vertical lines, respectively, contained in them.We investigate abstract counterparts of such sets of circles and partitions in arbitrary 2-dimensional Laguerre planes. We also prove a number of related results for generalized quadrangles associated with 2-dimensional Laguerre planes.

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