In (polyhedral)geometry, aflag is a sequence offaces of apolytope, each contained in the next, with exactly one face from eachdimension.
More formally, aflagψ of ann-polytope is a set{F–1,F0, ...,Fn} such thatFi ≤Fi+1(–1 ≤i ≤n – 1) and there is precisely oneFi inψ for eachi,(–1 ≤i ≤n). Since, however, the minimal faceF–1 and the maximal faceFn must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are calledimproper faces.
For example, a flag of apolyhedron comprises onevertex, oneedge incident to that vertex, and onepolygonal face incident to both, plus the two improper faces.
A polytope may be regarded as regular if, and only if, itssymmetry group istransitive on its flags. This definition excludeschiral polytopes.
In the more abstract setting ofincidence geometry, which is a set having a symmetric and reflexiverelation calledincidence defined on its elements, aflag is a set of elements that are mutually incident.[1] This level of abstraction generalizes both the polyhedral concept given above as well as the relatedflag concept from linear algebra.
A flag ismaximal if it is not contained in a larger flag. An incidence geometry (Ω,I) hasrankr if Ω can be partitioned into sets Ω1, Ω2, ..., Ωr, such that each maximal flag of the geometry intersects each of these sets in exactly one element. In this case, the elements of set Ωj are called elements oftypej.
Consequently, in a geometry of rankr, each maximal flag has exactlyr elements.
An incidence geometry of rank 2 is commonly called anincidence structure with elements of type 1 called points and elements of type 2 called blocks (or lines in some situations).[2] More formally,