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 ( r n , 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 ( r n , 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 S , we also show how to find a BEL-configuration corresponding to the dual spread S ϵ . Furthermore, we study the effect of polarities in V ( r n , 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 ( 2 n , q ) introduced in Ball et al. ( 2007 ), which, together with the transpose operation, leads to a group of order 8 acting on BEL-configurations.