Journal of Geometry

, Volume 105, Issue 3, pp 577–599 | Cite as

Planar nearrings on the Euclidean plane

  • Wen-Fong Ke
  • Hubert KiechleEmail author
  • Günter Pilz
  • Gerhard Wendt


Planar near-rings are generalized rings which can serve as coordinate domains for geometric structures in which each pair of nonparallel lines has a unique point of intersection. It is known that all planar nearrings can be constructed from regular groups of automorphisms of groups which can be viewed as the “action groups” of the planar nearring. In this article, we study planar nearrings whose additive group is \({(\mathbb{R}^n,+)}\), in particular, n = 1 and 2. It is natural to study topological planar nearrings in this context, following ideas of the late Kenneth D. Magill, Jr. In the case of n = 1, we characterize all topological planar nearrings by their action groups \({(\mathbb{R}^*, \cdot)}\) or \({(\mathbb{R}^+, \cdot)}\). For n = 2, these action groups and the circle group \({(\mathbb{U}, \cdot)}\) seem to be the most interesting cases, but the last case can be excluded completely. As a consequence, we obtain characterizations of the semi-homogeneous continuous mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}}\) for n = 1 and 2. Such a mapping f enjoys the property that f(f(u)v) = f(u)f(v) for all \({u,v \in \mathbb{R}^n}\). When \({f(\mathbb{R}^n) = \mathbb{R}^+}\), f is a positive homogeneous mapping of degree 1.

Mathematics Subject Classification (2010)

Primary 16Y30 Secondary 16W80 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Wen-Fong Ke
    • 1
  • Hubert Kiechle
    • 2
    Email author
  • Günter Pilz
    • 3
  • Gerhard Wendt
    • 3
  1. 1.Department of Mathematics and National Center for Theoretical SciencesNational Cheng Kung UniversityTainanTaiwan
  2. 2.Fachbereich MathematikUniversität HamburgHamburgGermany
  3. 3.Department of AlgebraJohannes Kepler Universität LinzLinzAustria

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