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]:

\((IP)\) 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], 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.