Let Γ(X, * , t, I) be an incidence geometry. Given a type i∈I, for any flag F of Γ, we define the i-shadow σi(F) as the set of elements of type i incident with F.
We define the intersection property (IP) as it appears in [Bue79]:
(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 Γ satisfy (IP).
If Γ is a geometry of rank n, we could define a property (IP)k in the following way (for k = 2, ..., n) as suggested by Francis Buekenhout:
(IP)k for every residue R of rank k of Γ, 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 [JL04], Pascale Jacobs and Dimitri Leemans have designed good algorithms to test these intersection properties. The Magma implementation is available with the following functions.
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.
Given a coset geometry C, this function returns the boolean value true if and only if C satisfies the intersection property.
Given a coset geometry C, this function returns the boolean value true if and only if C satisfies the weak intersection property.