incidence structure (original) (raw)

Definition. An incidence structure 𝒮 is a triple (𝒫,ℬ,ℐ), where

1. 1. 𝒫 and ℬ are two disjoint sets; the elements of 𝒫 and ℬ are respectively points and blocks of 𝒮, and

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 𝒫B⊆𝒫 of points it is incident with. The collection of all 𝒫B is a multiset, since it is possible that identical sets of points be related to distinct blocks. When 𝒫B′≠𝒫B′′ whenever B′≠B′′, the incidence structure is said to be simple. In a simple incidence structure, we could identify each block B with its 𝒫B so that blocks no longer have sets of points they are incident with but are such sets. If we define it that way, then

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  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 ℬP⊆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.

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 𝒮1=(𝒫1,ℬ1,ℐ1), 𝒮2=(𝒫2,ℬ2,ℐ2), a homomorphism from 𝒮1 to 𝒮2 is a pair of functions f:𝒫1→𝒫2 and g:ℬ1→ℬ2 such that (P,B)∈ℐ1 iff (f⁢(P),g⁢(B))∈ℐ2. 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 𝒮1 and 𝒮2 are simple, then a homomorphism can be thought of as a single function f:𝒫1→𝒫2 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

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 q2 points and q2+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.

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  A (finite) projective plane is an S⁢(2,q+1,q2+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 q2+q+1. Much harder is determining for which qsuch 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(mod4), 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 (readhttp://www.cecm.sfu.ca/organics/papers/lam/index.htmlhttp://www.cecm.sfu.ca/organics/papers/lam/index.html for Lam’s account).

References

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• 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