# Incidence system

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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.

How to Cite This Entry:
Incidence system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Incidence_system&oldid=19258
This article was adapted from an original article by V.E. Tarakanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article