On the splitting case of a semi-biplane construction

TL;DR

This paper analyzes how a specific incidence structure can be split into two substructures, providing a complete description and showing limitations of the construction for certain semi-biplanes.

Contribution

It offers a detailed analysis of the splitting case in semi-biplane constructions and identifies cases where the construction does not apply.

Findings

Complete description of the two substructures obtained.

The semi-biplanes sbp(18,6) cannot be constructed using this method.

Insights into the limitations of the semi-biplane construction.

Abstract

We consider the case where a particular incidence structure splits into two substructures. The incidence structure in question was used previously by the authors to construct semi-biplanes $sbp(k^2,k)$ or $sbp(k^2/2,k)$. A complete description of the two substructures is obtained. We also show that none of the three semi-biplanes, $sbp(18,6)$, can be described using this construction.