Residues

Let \(\Gamma(X,\sim,t,I)\) 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 \(x \sim F\).

The residue \(\Gamma_F\) of the flag \(F\) in \(\Gamma\) is the geometry whose set of elements is \(\lbrace x\in X : x \sim F \rbrace \backslash F\) and whose set of types is \(I\backslash t(F)\), 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 \(\Gamma_F = \Gamma( \cap_{j\in F} G_j; (G_i\cap (\cap_{j\in F}G_j))_{i \in I\backslash t(F)})\).

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 \(f\).