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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.

Semiaffine spaces.

Van Maldeghem, Hendrik (2009)

The Electronic Journal of Combinatorics [electronic only]

Several observations about Maneeals - a peculiar system of lines

Naga Vijay Krishna Dasari, Jakub Kabat (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

For an arbitrary triangle ABC and an integer n we define points Dn, En, Fn on the sides BC, CA, AB respectively, in such a manner that |AC|n|AB|n=|CDn||BDn|,|AB|n|BC|n=|AEn||CEn|,|BC|n|AC|n=|BFn||AFn|. A C n A B n = C D n B D n , A B n B C n = A E n C E n , B C n A C n = B F n A F n . Cevians ADn, BEn, CFn are said to be the Maneeals of order n. In this paper we discuss some properties of the Maneeals and related objects.

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