A $(0,1)$ — Matrix Framework for Elliptic Semiplanes

Abstract

We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types $C,D$, and $L$. Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach applies also to the only elliptic semiplane of type B known so far. In particular, the constructions provide intrinsic tactical decompositions and partitions for these elliptic semiplanes into elliptic semiplanes of smaller order.