Incidence structure

In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which points are on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane. Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects (planes, solids, n-spaces, conics, etc.) can be used. The study of finite structures is sometimes called finite geometry. An incidence structure is a triple (P, L, I) where P is a set whose elements are called points, L is a distinct set whose elements are called lines and I ⊆ P × L is the incidence relation. The elements of I are called flags. If (p, l) is in I then one may say that point p "lies on" line l or that the line l "passes through" point p. A more "symmetric" terminology, to reflect the symmetric nature of this relation, is that "p is incident with l" or that "l is incident with p" and uses the notation p I l synonymously with (p, l) ∈ I. In some common situations L may be a set of subsets of P in which case incidence I will be containment (p I l if and only if p is a member of l). Incidence structures of this type are called set-theoretic. This is not always the case, for example, if P is a set of vectors and L a set of square matrices, we may define I = {(v, M) : vector v is an eigenvector of matrix M }. This example also shows that while the geometric language of points and lines is used, the object types need not be these geometric objects. Incidence geometry An incidence structure is uniform if each line is incident with the same number of points.