incidence structure

incidence structure

Definition

. An

incidence structure

𝒮

is a triple

(

𝒫

,

ℬ

,

ℐ

)

, where

1.

𝒫

and

ℬ

are two

disjoint sets

; the elements of

𝒫

and

ℬ

are respectively

points

and

blocks

of

𝒮

, and

2.

ℐ

⊆

𝒫

×

ℬ

called the

incidence relation

of

𝒮

.

and a point

P

and a block

B

are said to be

incident

iff

(

P

,

B

)

∈

ℐ

. The

dual

incidence structure

ℐ

*

is the same

structure

with the labels “point” and “block” reversed.

Every block

B

has a set

𝒫

B

⊆

𝒫

of points it is incident with. The

collection

of all

𝒫

B

is a

multiset

, since it is possible that identical sets of points be related to distinct blocks. When

𝒫

B

′

≠

𝒫

B

′′

whenever

B

′

≠

B

′′

, the incidence structure is said to be

simple

. In a simple incidence structure, we could identify each block

B

with its

𝒫

B

so that blocks no longer

have

sets of points they are incident with but

are

such sets. If we define it that way, then

•

a

simple

incidence structure consists of a set

𝒫

and a set

ℬ

⊆

P

⁢

(

𝒫

)

where

P

⁢

(

𝒫

)

is the

powerset

of

𝒫

.

A simple incidence structure is also called a

hypergraph

(with points as vertices, and blocks as an extended type of “edges” that are no longer restricted to exactly two vertices each).

Every point

P

also has a set

ℬ

P

⊆

ℬ

of blocks it is incident with. Often, a simple incidence structure also has a simple dual, but the

set theory

formalism

does not allow us to regard blocks as sets of points and simultaneously points as sets of blocks! Nevertheless, it is often useful to alternate between these dual

interpretations

.

The definition given above is quite general, so one can easily come up with arbitrary examples. Nevertheless, interesting examples of incidence structures are found mainly in

geometry

and

combinatorics

. In geometry, incidence is usually interpreted as

set inclusion

, so when we say a line is incident with a plane, we are saying that the line is included (as a subset) in the plane. In combinatorics, the main use of incidence structure is in the study of

block designs

: grouping a finite collection of objects so that certain “incidence” properties are satisfied.

Incidence structures are examples of

relational structures

. As such, we can define

substructures

and

homomorphisms

between structures:

Definition

. Given an incidence structure

𝒮

=

(

𝒫

,

ℬ

,

ℐ

)

, a

substructure

of

𝒮

is an incidence structure

(

𝒫

′

,

ℬ

′

,

ℐ

′

)

such that

𝒫

′

⊆

𝒫

,

ℬ

′

⊆

ℬ

, and

ℐ

′

=

ℐ

∩

(

𝒫

′

×

ℬ

′

)

.

Definition

. Given two incidence structures

𝒮

1

=

(

𝒫

1

,

ℬ

1

,

ℐ

1

)

,

𝒮

2

=

(

𝒫

2

,

ℬ

2

,

ℐ

2

)

, a

homomorphism

from

𝒮

1

to

𝒮

2

is a pair of

functions

f

:

𝒫

1

→

𝒫

2

and

g

:

ℬ

1

→

ℬ

2

such that

(

P

,

B

)

∈

ℐ

1

iff

(

f

⁢

(

P

)

,

g

⁢

(

B

)

)

∈

ℐ

2

. A homomorphism is an

isomorphism

if both

f

and

g

are

bijections

. An isomorphism is an

automorphism

if

𝒮

′

=

𝒮

. It is easy to see that if both

𝒮

1

and

𝒮

2

are simple, then a homomorphism can be thought of as a single function

f

:

𝒫

1

→

𝒫

2

such that

P

∈

B

iff

f

⁢

(

P

)

∈

f

⁢

(

B

)

, where

f

⁢

(

B

)

=

{

f

⁢

(

Q

)

∣

Q

∈

B

}

.

Incidence structures are special cases of a general form of geometry called

Buekenhout-Tits geometry

. Given an incidence structure

(

𝒫

,

ℬ

,

ℐ

)

, form the

disjoint union

Λ

of

𝒫

and

ℬ

, and define a function

τ

:

Λ

→

{

0

,

1

}

where

τ

⁢

(

x

)

=

0

iff

x

is a point. Finally, define

binary relation

#

on

Λ

so that

x

⁢

#

⁢

y

iff one is incident with another, or

x

=

y

. Then

(

Λ

,

#

,

{

0

,

1

}

,

τ

)

so constructed is a geometry of rank 2.

Finite planes

The term

line

has a specific meaning for 2-designs in general: for any two points, it is the

intersection

of all blocks containing both those points. For 2-designs that are also

Steiner systems

(

τ

=

2

and

λ

=

1

) there
is only one such block, so line becomes a synonym for block. And it becomes a finite analogue of the usual geometric meaning of the word.

•

An

S

⁢

(

2

,

κ

,

ν

)

is the finite analogue of a

plane

, with blocks in the rôle of

lines

in the following sense: the design property now requires there to be, for any two different points, exactly one line “through” both those points. Just like in a real (

continuous

) plane.

This also implies that, for any two different lines

l

and

m

, there is

no more

than one point “on” both those lines (if both of

P

and

Q

were on both those lines, there would be two lines through those points).
It does not imply there is always such a point: just like in a real plane, lines can be

parallel

.

One example is a (finite)

affine plane

with

q

2

points and

q

2

+

q

lines. It can be obtained by deleting one line (and all its points) from a

projective plane

(for which see below). Lines that used to intersect in one of the deleted points are parallel in the affine plane.

•

A (finite)

projective plane

is an

S

⁢

(

2

,

q

+

1

,

q

2

+

q

+

1

)

and it has no

parallel lines

. Because any two lines meet in a point, the dual is again a projective plane. So a projective plane is a square design, as well as being a great many other things.

It is easy to prove that the property of being a plane dual to a plane (i.e. the absence of parallel lines) implies, apart from a few

trivial

cases, numbers of the form

q

+

1

and

q

2

+

q

+

1

. Much harder is determining for which

q

such planes exist. The parameter

q

is known as the

order

of the plane (this agrees with order as defined above for designs in general).

Highly

symmetric

“classical” (aka

Desarguesian

,

Pappian

) projective planes can be constructed based on finite fields, for any prime power

q

. Many non-Desarguesian projective planes are known, but thus far their

q

are also prime powers. The

prime power conjecture

is that orders of all projective planes will be prime powers.

The

Bruck–Ryser

theorem

states that if

q

≡

1

or

2

(

mod

4

)

, and not (a square or) the sum of two squares, it cannot be the order of a projective plane. This rules out 6 for

instance

, as well as 14 etc. It has been extended
to the

Bruck–Ryser–Chowla theorem

for all square 2-designs, with a more complicated constraint.

The only other order ruled out to date is 10, via an epic computer search by Lam, Swiercz and Thiel (read

http://www.cecm.sfu.ca/organics/papers/lam/index.htmlhttp://www.cecm.sfu.ca/organics/papers/lam/index.html

for Lam’s account).

References

1

AK93

E. F. Assmus

and

J. D. Key

,

Designs and their Codes

(pbk. ed. w. corr.), Camb. Univ. Pr. 1993,

ISBN  0 521 45839 0

first part has thorough

introduction

to
various flavors of incidence structure

Cam94

Peter J. Cameron

,

Combinatorics: topics, techniques, algorithms

,

Camb. Univ. Pr. 1994,

ISBN  0 521 45761 0

http://www.maths.qmul.ac.uk/ pjc/comb/http://www.maths.qmul.ac.uk/ pjc/comb/

(

solutions

, errata &c.)

good combinatorics textbook, with detail

Pot95

Alexander Pott

,

Finite Geometry

and Character Theory

,

Lect. Notes in Math.

1601

, Springer 1995,

ISBN  3 540 59065 X

includes clear introduction to incidence
structures

CD96

Charles J. Colbourn

and

Jeffrey H. Dinitz

, eds.

The CRC Handbook of

Combinatorial Designs

,

CRC Press 1996,

ISBN  0 8493 8948 8

http://www.emba.uvm.edu/ dinitz/hcd.htmlhttp://www.emba.uvm.edu/ dinitz/hcd.html

(errata, new results)

the reference work on designs incl. Steiner systems,
proj. planes

Title

incidence structure

Canonical name

IncidenceStructure

Date of creation

2013-03-22 15:10:56

Last modified on

2013-03-22 15:10:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Topic

Classification

msc 62K10

Classification

msc 51E30

Classification

msc 51E05

Classification

msc 05B25

Classification

msc 05B07

Classification

msc 05B05

Related topic

Hypergraph

Related topic

SteinerSystem

Related topic

TacticalDecomposition

Related topic

ProjectivePlane2

Related topic

FiniteProjectivePlane4

Related topic

BuekenhoutTitsGeometry

Defines

incidence relation

Defines

point

Defines

block

Defines

incident

Defines

simple incidence structure

Defines

affine plane

Defines

finite affine plane

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