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A Class of Homomorphisms of Pre-Hjelmslev Groups

Published online by Cambridge University Press:  20 November 2018

Frieder Knüppel
Frieder Knüppel*
Affiliation:
Universität Kiel, Kiel, W. Germany
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E. Salow [8] introduced the concept of pre-Hjelmslev groups, a generalization of F. Bachmann's Hjelmslev groups [1] which leads to a more natural theory of homomorphisms and permits a simpler construction of algebraic models. Basically, both types of groups are the groups of motions of a metric plane, the so-called group plane. In such a plane there is a unique perpendicular through any point to any line and the product of three collinear points (three copunctal lines) is a point (a line). Our first section contains the precise definitions and some basic facts.

The homomorphic image of a pre-Hjelmslev group can be more complicated than the pre-image. For instance, there may always be a unique line through two distinct points of the pre-image but not of the image. We study regular homomorphisms of pre-Hjelmslev groups, i.e., homomorphisms with the following property: If two lines intersect at exactly one point, their images will also have precisely one point in common.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 3 , 01 June 1984 , pp. 470 - 494
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Bachmann, F., Aufbau der Geometrie aus clem Spiege lungs be griff, Second supplemented edition (Springer, 1973).CrossRefGoogle Scholar
2. Bachmann, F., Hjelmslev-Gruppen. Arbeitsgemeinschaft über geometrische Fragen, Universität Kiel, second print (1976).Google Scholar
3. Klingenberg, W., Euklidische Ebenen mit Nachbar element en, Math. Z. 61 (1954), 1–25.Google Scholar
4. Knüppel, F. and Kunze, M., Neighbor relation and neighbor homomorphism of Hjelmslev groups, Can. J. Math. 31 (1979), 680–699.Google Scholar
5. Knüppel, F. and Kunze, M., Regulare Hjelmslev-Homomorphismen, Geom. Ded. 11 (1981), 195–225.Google Scholar
6. Knüppel, F., Äquiforme Ebenen über kommutativen Ringen und singuläre Prä-Hjelmslev-gruppen, Abh. math. Sem. Univ. Hamburg 54 (1983), 229–257.Google Scholar
7. Knüppel, F., The set of fixed points of a rotation in a pre-Hjelmslev group, to appear.Google Scholar
8. Salow, E., Singuläre Hjelmslev-Gruppen, Geom. Ded. 1 (1973), 447–467.Google Scholar
9. Salow, E., Fixpunktmenge von Drehungen in Hjelmslev-Gruppen, Abh. math. Sem. Univ. Hamburgh (1974), 37–73.Google Scholar