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.
More formally, a flag of an -polytope is a set such that and there is precisely one in for each , Since, however, the minimal face and the maximal face 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 -adjacent if they only differ by a face of rank . They are adjacent if they are -adjacent for some value of . Each flag is -adjacent to precisely one flag.
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 (é, ) has rank if é can be partitioned into sets é<sub>1</sub>, é<sub>2</sub>, ..., é<sub></sub>, such that each maximal flag of the geometry intersects each of these sets in exactly one element. In this case, the elements of set é<sub></sub> 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 (or lines in some situations). More formally,