TY - JOUR

T1 - Planar nearrings on the Euclidean plane

AU - Ke, Wen-Fong

AU - Kiechle, Hubert

AU - Pilz, Günter

AU - Wendt, Gerhard

PY - 2014/10/10

Y1 - 2014/10/10

N2 - 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 (Rn, +), 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 ( R*,.)or( R∗,.). For n = 2, these action groups and the circle group ( U,.) 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 Rn to 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 ∈ Rn. When (Rn) R+, f is a positive homogeneous mapping of degree 1.

AB - 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 (Rn, +), 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 ( R*,.)or( R∗,.). For n = 2, these action groups and the circle group ( U,.) 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 Rn to 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 ∈ Rn. When (Rn) R+, f is a positive homogeneous mapping of degree 1.

UR - http://www.scopus.com/inward/record.url?scp=84939873083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939873083&partnerID=8YFLogxK

U2 - 10.1007/s00022-014-0221-7

DO - 10.1007/s00022-014-0221-7

M3 - Article

AN - SCOPUS:84939873083

VL - 105

SP - 577

EP - 599

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 3

ER -