A family of two sets and with an incidence relation between their elements, which is written as for , . In this case one says that the element is incident with , or that is incident with . 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 and are called, respectively, points and lines, and 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 (cf. Block design), which are obtained by requiring that 1) each is incident with precisely elements of ; 2) each is incident with precisely elements of ; and 3) each pair of distinct elements of is incident with precisely elements of . Often a set of subsets of is taken for ; then is simply .
Two incidence systems and are called isomorphic if there are one-to-one correspondences and such that
If and are finite sets, then the properties of the incidence system can be conveniently described by the incidence matrix , where if , and otherwise. The matrix determines up to an isomorphism.
|||M. Hall, "Combinatorial theory" , Blaisdell (1967)|
|||R. Dembowski, "Finite geometries" , Springer (1968)|
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 and 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.
|[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)|
Incidence system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Incidence_system&oldid=19258