On BEL-configurations and finite semifields

The BEL-construction for finite semifields was introduced in Ball et al. (J Algebra 311:117–129, 2007); a geometric method for constructing semifield spreads, using so-called BEL-configurations in \(V(rn,q)\). In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in \(V(rn,q)\), extending the results from Ball et al. (2007), where this was obtained only for \(r=n\). Given a BEL-configuration with associated semifield spread \(\mathcal S\), we also show how to find a BEL-configuration corresponding to the dual spread \(\mathcal {S}^{\epsilon }\). Furthermore, we study the effect of polarities in \(V(rn,q)\) on BEL-configurations, leading to a characterisation of BEL-configurations associated to symplectic semifields. We give precise conditions for when two BEL-configurations in \(V(n^2,q)\) define isotopic semifields. We define operations which preserve the BEL property, and show how non-isotopic semifields can be equivalent under this operation. We also define an extension of the “switching” operation on BEL-configurations in \(V(2n,q)\) introduced in Ball et al. (2007), which, together with the transpose operation, leads to a group of order \(8\) acting on BEL-configurations.