Flag (geometry)


In geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.
More formally, a flag ψ of an n-polytope is a set such that Fi ≤ Fi+1 and there is precisely one Fi in ψ for each i,. Since, however, the minimal face F−1 and the maximal face Fn must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces.
For example, a flag of a polyhedron comprises one vertex, one edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces.
A polytope may be regarded as regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.

Incidence geometry

In the more abstract setting of incidence geometry, which is a set having a symmetric and reflexive relation called incidence defined on its elements, a flag is a set of elements that are mutually incident. This level of abstraction generalizes both the polyhedral concept given above as well as the related flag concept from linear algebra.
A flag is maximal if it is not contained in a larger flag. An incidence geometry has rank if Ω can be partitioned into sets Ω1, Ω2,..., Ω, such that each maximal flag of the geometry intersects each of these sets in exactly one element. In this case, the elements of set Ω are called elements of type .
Consequently, in a geometry of rank, each maximal flag has exactly elements.
An incidence geometry of rank 2 is commonly called an incidence structure with elements of type 1 called points and elements of type 2 called blocks. More formally,