BEL-构形与有限半域
On BEL-configurations and finite semifields
摘要：
在\cite{BEL2007}中引入了有限半域的BEL-构造;一种构造半域扩展的几何方法，使用$V(rn,q)$中的所谓BEL-配置。在本文中，我们研究这种结构更详细，并确定了一个明确的乘法与一个贝尔配置在$V(rn,q)$的半域，从\cite{BEL2007}，其中这是得到的结果，只有$r=n$。给出了一个BEL-配置与相关的半域扩展$\mathcal{S}$，我们还展示了如何找到一个BEL-配置对应的双重扩展$\mathcal{S}^d$。此外，我们还研究了$V(rn,q)$的极性对BEL-组态的影响，从而得到了与辛半域相关的BEL-组态的特征。给出了$V(n^2,q)$中两种BEL组态定义同位素半场的精确条件。我们定义的操作，保持BEL属性，并显示如何在此操作下的非同位素半域可以是等同的。我们还定义了\cite{BEL2007}中引入的$V(2n,q)$中BEL-组态上的“开关”操作的扩展，它与转置操作一起导致一组作用于BEL-组态的序$8$.
英文摘要：
The BEL-construction for finite semifields was introduced in \cite{BEL2007}; 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 \cite{BEL2007}, where this was obtained only for $r=n$. Given a BEL-configuration with associated semifields spread $\mathcal{S}$, we also show how to find a BEL-configuration corresponding to the dual spread $\mathcal{S}^d$. 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 \cite{BEL2007}, which, together with the transpose operation, leads to a group of order $8$ acting on BEL-configurations.
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