Incidence Geometry

Magma 2.7 contains facilities for creating and computing with incidence geometries and coset geometries. These have been developed by Dimitri Leemans (Brussels). The Magma Incidence Structure type comprises a set of points and a set of blocks together with an incidence relation. Following Bekenhout, we define a more general object as follows: An incidence geometry is a 4-tuple $\Gamma = (X, *, t, I)$ where

• X is a set of elements;
• I is a non-empty set whose elements are called types;
• $t : X \rightarrow I : x \mapsto t(x)$ is a type function which maps an element to its type;
• * is an incidence relation that is a reflexive and symmetric relation such that $\forall x,y \in X$, x * y and t(x) * t(y) $\Rightarrow x = y$.

We also introduce group-geometry pairs or coset geometries. Roughly speaking, these are geometries constructed from a group and some of its subgroups in the following way. Let I be a finite set and let G be a group together with a finite family of subgroups (G_i) $_{i \in I}$. We define the incidence geometry $\Gamma = \Gamma(G,(G_i)_{i \in I})$ as follows. The set X of elements or varieties of $\Gamma$ consists of all cosets gG_i, $g \in G$, $i \in I$. We define an incidence relation * on X by:

g₁G_i * g₂G_j iff $g_1G_i \cap g_2G_j$ is non-empty in G.

$\Gamma(G,(G_i)_{i \in I})$ may also be called a group-geometry pair.