Incidence system
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
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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.
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 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.
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) |
Incidence system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Incidence_system&oldid=37004