Residues

Let Gamma(X, ~, t, I) be an incidence geometry and let F be a flag of Gamma (i.e. a clique of the incidence graph of Gamma).

We say that an element x in F is incident to the flag F if and only if x is incident to all elements in F, and we denote it x ~F.

The residue Gamma_F of the flag F in Gamma is the geometry whose set of elements is { x in X : x ~F } \ F and whose set of types is I\ t(F), together with the restricted type function and incidence relation.

Let Gamma(G;(G_i)_(i in I)) be a coset geometry and assume that G acts flag--transitively on Gamma. Let F be a flag of Gamma. The residue of F is the coset geometry Gamma_F = Gamma( cap_(j in F) G_j; (G_iintersect (cap_(j in F)G_j))_(i in I\ t(F))).

Residue(D, f) : IncGeom, Set -> IncGeom

Given an incidence geometry D and a flag f of D, return the residue of the flag f as an incidence geometry.

Residue(C, f) : CosetGeom, Set -> CosetGeom

Given a coset geometry C and a subset f of the set of types of C, return the residue of the flag consisting in the maximal parabolics of C whose type is in f.