Circular planar nearrings: geometrical and combinatorial aspects

TL;DR

This paper explores the geometric and combinatorial properties of disks in circular Ferrero pairs, revealing analogies with Euclidean geometry and conditions under which these structures form balanced incomplete block designs.

Contribution

It introduces a new definition of disks in circular Ferrero pairs and demonstrates their geometric properties and conditions for forming balanced incomplete block designs.

Findings

Many analogies with Euclidean geometry in the field-generated case

Conditions under which the incidence structure forms a balanced incomplete block design

Identification of interesting cases where these properties hold

Abstract

Let $(N,\Phi)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\mathcal{D}(a;b)$, as \[\mathcal{D}(a;b)=\{x\in \Phi(r)+c\mid r\neq 0,\ b\in \Phi(r)+c,\ |(\Phi(r)+c)\cap (\Phi(a)+b)|=1\}.\] 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.