Incidence system

Definition
An incidence system $$(G,*,t,I)$$ is the following data:


 * A set of elements $$G$$
 * A reflexive symmetric binary relation $$*$$ called the incidence relation
 * A set $$I$$ of possible types
 * A type function $$t:G \to I$$ which essentially partitions the elements of $$G$$ into types, such that no two elements of the same type are incident on each other

In the language of graph theory, an incidence system can be thought of as a $$|I|$$-partite graph structure with vertex set $$G$$ and the various parts being $$t^{-1}(i)$$ for $$i \in I$$.

Rank of an incidence system
The rank of an incidence system is defined as the cardinality of the image of $$t$$. This roughly corresponds to the notion of chromatic number in graph theory.

Flag
A flag is a set of pairwise incident elements. This corresponds to a clique in the corresponding graph.

The type of a flag is the set of types of all elements in it, and the rank of a flag is the number of elements in it.

The cotype is the complement of the type in the set $$I$$ and the corank is the size of the cotype.