Combinatorial properties of ring generated circular planar nearrings
A planar near-ring \((N,+,\cdot)\), its corresponding Ferrero pair \((N,\Phi)\) and the incidence structure \((N,{\mathcal B}^*_\Phi,\varepsilon)\), where \[ {\mathcal B}^*_\Phi=\bigl\{ N^*a+b \mid a, b \in N,\;a \neq 0 \bigr\}=\bigl\{ \Phi (a)=b \mid a, b \in N,\;a \neq 0 \bigr\}, \] and \(N^*=\{n \in N\mid n \not \equiv_m 0\}\) and \(\Phi(a)=\{\varphi(a)\mid\varphi\in\Phi\}\), are said to be circular, if any three distinct points of \(N\) belong to at most one block \(N^*_{a+b}\). Let \(N\) be a ring and \(\Phi=\langle varphi\rangle\) be a cyclic subgroup of \(U(N)\), of order \(k\), such that \((N,\Phi)\) is a circular Ferrero pair. For \(r,c \in N\setminus\{0\}\), define \(E^r_c=\{\Phi r+b\mid b\in\Phi_c\}\), a circle with radius \(r\) and center on the circle \(\Phi_c\). The authors describe the connection between an \(E^r_c\) and its corresponding graph, showing that: 1. Each graph can be decomposed into a union of some basic subgraphs; 2. The total number of basic graphs occurring depends only on \(k\); 3. For field generated planar near-rings, some graphs always occur together as subgroups of some \(E^r_c\) regardless of the underlying field.
scientific article; zbMATH DE number 842852