Skip to main content
Log in

A classification of Minkowski planes over half-ordered fields

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

This paper concerns a construction of Minkowski planes over half-ordered fields [5] and [20]. Solving various functional equations the Klein-Kroll types of these Minkowski planes are determined with respect toG- andq-translations and (p, q)-homotheties. Examples for some of the resulting types are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Carlitz, L.: A theorem on permutations in a finite field. Proc. Amer. Math. Soc. 11 (1960), 456–459

    Google Scholar 

  2. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York, 1969

    Google Scholar 

  3. Jakóbowski, J.: Conjugate points of a translation plane of order 9 and its extensibility to a Minkowski plane. Zeszyty Nauk. Geom. 17 (1988), 85–96

    Google Scholar 

  4. Jakóbowski, J.M.: A new generalization of Moulton affine planes. Geom. Dedicata 42 (1992), 243–253

    Google Scholar 

  5. Jakóbowski, J.M.: A full generalization of Moulton's method applied to Minkowski axioms. Zeszyty Nauk. Geom. 20 (1993), 13–21

    Google Scholar 

  6. Jakóbowski, J.M.: A new construction for Minkowski planes. Geom. Dedicata 69 (1998), 179–188

    Google Scholar 

  7. Jakóbowski, J.M.,Kroll, H.-J. andMatraś, A.: Minkowski planes admitting automorphism groups of small type. Preprint

  8. Klein, M.: Classification of Minkowski planes by transitive groups of homotheties. J. Geom. 43 (1992), 116–128

    Google Scholar 

  9. Klein, M. andKroll, H.-J.: A classification of Minkowski planes. J. Geom. 36 (1989), 99–109

    Google Scholar 

  10. Klein, M. andKroll, H.-J.: On Minkowski planes with transitive groups of homotheties. Abh. Math. Sem. Univ. Hamburg 64 (1994), 303–313

    Google Scholar 

  11. Kroll, H.-J.: Eine Kennzeichnung der miquelschen Minkowski-Ebenen durch Transitivitätseigenschaften. Geom. Dedicata 58 (1995), 75–77

    Google Scholar 

  12. Kroll, H.-J. andMatraś, A.: Minkowski planes with miquelian pairs. Beitr. Algebra Geom. 38 (1997), 99–109

    Google Scholar 

  13. Lang, S.: Algebra. 7th ed., Addison-Wesley, Reading, Massachusetts 1977

    Google Scholar 

  14. Meschiar, I.M. andQuattrocchi, P.: Una classificazione delle strutture di incidenza associate a insiemi di sostituzioni strettamente 3-transitivi finiti. Atti Sem. Mat. Fis Univ. Modena 24 (1975), 123–141

    Google Scholar 

  15. Perscy, N.: Les plans de Minkowski possedant des inversions sont miquellien. Simon Stevin 57 (1983), 15–35

    Google Scholar 

  16. Steinke, G.F.: A remark on Benz planes of order 9. Ars Comb. 34 (1992), 257–267

    Google Scholar 

  17. Steinke, G.F.: A family of 2-dimensional Minkowski planes with small automorphism groups. Result. Math. 26 (1994), 131–142

    Google Scholar 

  18. Steinke, G.F.: Topological circle geometries. in: Handbook of Incidence Geometry (ed. F. Buekenhout) Elsevier 1995, pp. 1327–1354

  19. Steinke, G.F.: Semiclassical projective planes over half-ordered fields. Geom. Dedicata 58 (1995), 21–44

    Google Scholar 

  20. Steinke, G.F.: A family of Minkowski planes over half-ordered fields. Beitr. Algebra Geom. 37 (1996), 355–366

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Steinke, G.F. A classification of Minkowski planes over half-ordered fields. J Geom 69, 192–214 (2000). https://doi.org/10.1007/BF01237486

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01237486

1991 Mathematics subject classification

Navigation