Incidence system

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This term is related to: incidence geometry
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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.

Terminology

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.