Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-06-08T22:01:45.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Neighbor Relation and Neighbor Homomorphism of Hjelmslev Groups

Published online by Cambridge University Press:  20 November 2018

Frieder Knüppel  and
Michael Kunze
Frieder Knüppel
Affiliation:
Universität Kiel, Kiel, W. Germany
Michael Kunze
Affiliation:
Technische Hochschule Darmstadt, Darmstadt, W. Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The geometry of Hjelmslev groups is a comprehensive plane metric geometry. It supplies, for example, an approach to euclidean, hyperbolic, elliptic, Minkowskian and Galilean geometry.

The subject of this geometry are Hjelmslev groups and their group planes. (The definition of Hjelmslev groups and some basic concepts and propositions of this theory can be found in the second edition of Bachmann's book [1] (pages 318-328). A similar report in English is the lecture [3]. A comprehensive introduction is developed in the key work [2]. The first part of this work was translated by Garner [4]. An abstract is given in Math. Rev. 52, 9066 (1976).) The group plane arises from giving geometric names to some group theoretical facts. The group plane of a Hjelmslev group is an incidence structure with orthogonality, and the Hjelmslev group acts on this plane as a group of motions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 680 - 699
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bachmann, F., Aufbau der Géométrie aus dem Spiegelungsbegriff, Second revised edition (New York, Heidelberg, Berlin: Springer, 1973).Google Scholar
2. Bachmann, F., Hjelmslev-Gruppen, Mathematisches Seminar der Universitat Kiel, 1970/71, Reprint (1976).Google Scholar
3. Bachmann, F., Hjelmslev groups, Atti del Colloquio Internazionale sulle Teorie combinatorie (Roma 1973), 469–479.Google Scholar
4. Bachmann, F., Rudiments of the theory of Hjelmslev groups, Part I of [2], translated by C.W.L. Garner, Toronto, 1974.Google Scholar
5. Benda, H. V. und Kniippel, F., Hjelmslev-Gruppen ilber lokalen Ringen, Geometriae Dedicat. 5 (1976), 195–206.Google Scholar
6. Hjelmslev, J., Einleitung in die allgemeine Kongruenzlehre, Danske Vid. Selsk., mat.-fys. Medd. 8, No. 11 (1929); 10, No. 1 (1929); 19, No. 12 (1942); 22, No. 6 (1945); 22, No. 13 (1945); 25, No. 10 (1949).Google Scholar
7. Klingenberg, W., Euklidische Ebenen mit Nachbarelementen, Math. Z. 61 (1954), 1–25.Google Scholar
8. Knùppel, F. und Kunze, M., Nachbar-Relation und Gross-Geometrie in Hjelmslev-Gruppen mit Sternaxiom, Mathematisches Seminar der Universitàt Kiel (1976).Google Scholar
9. Kunze, M., Angeordnete Hjelmslevsche Géométrie, Diss. Kiel (1975).Google Scholar
10. Salow, E., Singuldre Hjelmslev-Gruppen, Geometriae Dedicat. 1 (1973), 447–467.Google Scholar
11. Salow, E., Einbettung von Hjelmslev-Gruppen in orthogonale Gruppen ilber kommutativen Ringen, Math. Z. 134 (1973), 143–170.Google Scholar
12. Salow, E., Fixpunktmengen von Drehungen in Hjelmslev-Gruppen, Abh. math. Sem. Univ. Hambur. 41 (1974), 37–73.Google Scholar
13. Schnabel, R., Erweiterung und Faktorisierung von Hjelmslev-Gruppen , Diss. Kiel (1974).Google Scholar