# Intersection Properties of Coset Geometries

Let $\Gamma(X,*,t,I)$ be an incidence geometry. Given a type $i\in I$, for any flag $F$ of $\Gamma$, we define the $i$*-shadow* $\sigma_i(F)$ as the set of elements of type $i$ incident with $F$.

We define the intersection property $(IP)$ as it appears in [[Buekenhout, 1979](../../references.md#cite-buekenhout1979)]:

$(IP)$ *for every type* $i$*, the intersection of the* $i$*-shadows of a variety* $x$ *and a flag* $F$ *is empty or it is the* $i$*-shadow of a flag incident to* $x$ *and* $F$*. The same holds on the residues.*

In earlier works, the second author, together with other persons, among whom are Francis Buekenhout, Michel Dehon and Philippe Cara, imposed a condition called $(IP)_2$. This condition asks that all rank two residues of $\Gamma$ satisfy $(IP)$.

If $\Gamma$ is a geometry of rank $n$, we could define a property $(IP)_k$ in the following way (for $k = 2, \ldots, n$) as suggested by Francis Buekenhout:

$(IP)_k$ *for every residue* $R$ *of rank* $k$ *of* $\Gamma$*, for every type* $i$ *in the set of types of* $R$*, the intersection of the* $i$*-shadows of a variety* $x$ *and a flag* $F$ *is empty or it is the* $i$*-shadow of a flag incident to* $x$ *and* $F$*.*

In [[Jacobs and Leemans, 2004](../../references.md#cite-jl2004)], Pascale Jacobs and Dimitri Leemans have designed good algorithms to test these intersection properties. The Magma implementation is available with the following functions.

## `HasIntersectionPropertyN(C, n): CosetGeom, RngIntElt -> BoolElt, BoolElt`

Given a coset geometry $C$ and a positive integer $n$, this function determines firstly, whether $C$ satisfies the intersection property of rank $n$, and secondly, whether $C$ satisfies the weak intersection property of rank $n$. A boolean value corresponding to each of these cases is returned.

## `HasIntersectionProperty(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ satisfies the intersection property.

## `HasWeakIntersectionProperty(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ satisfies the weak intersection property.