Incidence system

A family $S = (A,\mathfrak{B},I)$ of two sets $A$ and $\mathfrak{B}$ with an incidence relation $I$ between their elements, which is written as $a\,I\,B$ for $a \in A$, $B \in \mathfrak{B}$. In this case one says that the element $a$ is incident with $B$, or that $B$ is incident with $a$. The concept of an incidence system is introduced with the purpose of using the language of geometry in the study of general combinatorial existence and construction problems; the incidence relation is ascribed certain properties that lead to some or other combinatorial configurations.

An example of incidence systems used in combinatorics are (finite) geometries: the elements of the (finite) sets $A$ and $\mathfrak{B}$ are called, respectively, points and lines, and $I$ is a relation with properties that are usual in the theory of projective or affine geometry. Another characteristic example of incidence systems is that of block designs, which are obtained by requiring that 1) each $a \in A$ is incident with precisely $r$ elements of $\mathfrak{B}$; 2) each $B \in \mathfrak{B}$ is incident with precisely $k$ elements of $A$; and 3) each pair of distinct elements of $A$ is incident with precisely $\lambda$ elements of $\mathfrak{B}$. Often a set of subsets of $A$ is taken for $\mathfrak{B}$; then $a\,I\,B$ is simply $a \in B$.

Two incidence systems $S = (A,\mathfrak{B},I)$ and $S' = (A',\mathfrak{B'},I')$ are called isomorphic if there are one-to-one correspondences $\alpha : A \leftrightarrow A'$ and $\beta : \mathfrak{B} \leftrightarrow \mathfrak{B'}$ such that $$ a\,I\,B \Leftrightarrow \alpha(a)\,I'\,\beta(B) \ . $$

If $A = \{a_1,\ldots,a_n\}$ and $\mathfrak{B} = \{B_1,\ldots,B_m\}$ are finite sets, then the properties of the incidence system $S$ can be conveniently described by the incidence matrix $\Sigma$, where $\Sigma_{ij} = 1$ if $a_i\,I\,B_j$, and $\Sigma_{ij} = 0$ otherwise. The matrix $\Sigma$ determines $S$ up to an isomorphism.

References

[1]	M. Hall, "Combinatorial theory" , Blaisdell (1967)
[2]	R. Dembowski, "Finite geometries" , Springer (1968)

Comments

Condition 1) for a block design follows from conditions 2) and 3).

A more general type of incidence system is a Buekenhout–Tits geometry, obtained when one considers not two sets $A$ and $\mathfrak{B}$ but infinitely many types of objects.

From the point of view of graph theory, an incidence system is a hypergraph.

An incidence system is also called an incidence structure.

References

[a1]	T. Beth, D. Jungnickel, H. Lenz, "Design theory" , B.I. Wissenschaftsverlag Mannheim (1985)
[a2]	A. Beutelspacher, "Einführung in die endliche Geometrie" , I-II , B.I. Wissenschaftsverlag Mannheim (1982–1983)