Incidence structure

In mathematics, an abstract system consisting of two types of objects and a single relationship between these types of objects is called an incidence structure. 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 disjoint set whose elements are called lines and $I \subseteq P \times L$ is the incidence relation. The elements of $I$ are called flags. If ( $p, l$ ) is in $I$ then it was typical to say that point $p$ "lies on" line $l$ or that the line $l$ "passes through" point $p$ . However, today a more "symmetric" terminology is preferred to reflect the symmetric nature of this relation, so one says that " $p$ is incident with $l$ " or that " $l$ is incident with $p$ " and uses the notation $p I l$ in lieu of $(p, l) \in I$ .

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