Residue

Residue is a concept in incidence systems that generalizes the concept of sections on abstract polytopes.

Definition

The residue of a flag, F, in an incidence geometry, $\Gamma = (X,*,t,I)$, is an incidence system $\Gamma_F = (X_F, *_F, t_F, I_F)$ where

• $X_F$ is the set of elements of Γ, not in F, that are incident on every element of F.
• $I_F$ is the set of types of $X_F$, $t(X_F)$. It is equivalent to $I\setminus t(F)$.
• $*_F$ is * restricted to the domain of $X_F$.
• $t_F$ is t restricted to the domain of $X_F$.

An incidence structure is residually connected if every residue of rank 2 or greater has a connected incidence graph.

Bibliography

• Fernandes, Maria (2014). "Regular and chiral hypertopes" (PDF).