Circular planar nearrings: geometrical and combinatorial aspects

Abstract

Let $(N,\Phi )$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\mathcal{D}(a;b)$, as \[\mathcal{D}(a;b)=\{x\in Φ(r)+c\mid r\neq 0,\ b\in Φ(r)+c,\ |(Φ(r)+c)\cap (Φ(a)+b)|=1\}.\] We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if ${\mathcal{B}}^{\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,{\mathcal{B}}^{\mathcal{D}},\in )$ is actually a balanced incomplete block design.

12 pages