Flag (geometry)

For a comparable but different concept from linear algebra, see flag (linear algebra).

In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.

Face diagram of a square pyramid showing one of its flags

More formally, a flag ψ of an n-polytope is a set {F_-1, F₀, ..., F_n} such that F_i ≤ F_i+1 (-1 ≤ i ≤ n – 1) and there is precisely one F_i in ψ for each i, (-1 ≤ i ≤ n). Since, however, the minimal face F_–1 and the maximal face F_n 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 is regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes.

Two flags are j-adjacent if they only differ by a face of rank j. They are adjacent if they are j-adjacent for some value of j. Each flag is j-adjacent to precisely one flag.^[1]

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.^[2] 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 (Ω, I) has rank r if Ω can be partitioned into sets Ω₁, Ω₂, ..., Ω_r, 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_j are called elements of type j.

Consequently, in a geometry of rank r, each maximal flag has exactly r 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 (or lines in some situations).^[3] More formally,

An incidence structure is a triple D = (V, B, I) where V and B are any two disjoint sets and I is a binary relation between V and B, that is, I ⊆ V × B. The elements of V will be called points, those of B blocks and those of I flags.^[4]