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A Construction of a Skewaffine Structure in Laguerre Geometry

Andrzej Matraś (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

J. Andre constructed a skewaffine structure as a group space of a normally transitive group. In this paper his construction is used to describe the structure of the set of circles not passing through a point of a Laguerre plane. Sufficient conditions to ensure that this structure is a skewaffine plane are given.

Central Automorphisms of Veblenian Nearaffine Planes

Kinga Cudna-Lasecka, Jan Jakóbowski (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

The paper deals with nearaffine planes described by H. A. Wilbrink. We consider their central automorphisms, i.e. automorphisms satisfying the Veblen condition, which become central collineations in connected projective planes. Moreover, a concept of central pseudo-automorphism is considered, i.e. some bijections in a nearaffine plane are not automorphisms but they become central collineations in the related projective planes.

Extending Nearaffine Planes to Hyperbola Structures

Kinga Cudna-Salmanowicz, Jan Jakóbowski (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

H. A. Wilbrink [Geom. Dedicata 12 (1982)] considered a class of Minkowski planes whose restrictions, called residual planes, are nearaffine planes. Our study goes in the opposite direction: what conditions on a nearaffine plane are necessary and sufficient to get an extension which is a hyperbola structure.

Some Generalization of Nearaffine Planes

Jan Jakóbowski (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

There are three kinds of Benz planes: Möbius planes, Laguerre planes and Minkowski planes. A Minkowski plane satisfying an additional axiom is connected with some other structure called a nearaffine plane. We construct an analogous structure for a Laguerre plane. Moreover, our description is common for both cases.

Strutture di André con gruppi di traslazioni transitivi non normali

Mauro Biliotti, Armin Herzer (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We give some examples of André-structures admitting translation groups which are transitive on the set of points but which are not normal in the dilatation group. André structures with this property seem to be new in the literature.

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