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Reflection loops of spaces with congruence and hyperbolic incidence structure

Alexander Kreuzer (2004)

Commentationes Mathematicae Universitatis Carolinae

In an absolute space ( P , 𝔏 , , α ) with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that ( P , 𝔏 , α ) is a linear space with hyperbolic incidence structure, the addition is not associative. ( P , + ) is a K-loop or a Bruck loop.

Some generalization of Desargues and Veronese configurations

Prazmowska, Malgorzata, Krzysztof, Prazmowski (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 51E14, 51E30.We propose a method of constructing partial Steiner triple system, which generalizes the representation of the Desargues configuration as a suitable completion of three Veblen configurations. Some classification of the resulting configurations is given and the automorphism groups of configurations of several types are determined.

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