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In [[mathematics]], an '''incidence structure''' is a triple
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:<math>C=(P,L,I).\,</math>
 
where ''P'' is a set of "points", ''L'' is a set of "lines" and <math>I \subseteq P \times L</math> is the [[Incidence (geometry)|incidence]] relation. The elements of <math>I</math> are called '''flags.''' If
:<math>(p,l) \in I,</math>
 
we say that point ''p'' "lies on" line <math>l</math>. One may concretely have ''L'' be a set of subsets of ''P,'' and have incidence ''I'' be containment (<math>(p,l) \in I</math> if and only if <math>p \in l</math>), but one may also work more abstractly.
 
Incidence structures generalize planes (such as [[affine plane (incidence geometry)|affine]], [[projective plane|projective]], and [[Möbius plane]]s) as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the finite structures are sometimes called [[Finite geometry|finite geometries]].
 
==Comparison with other structures==
An incidence figure (that is, a depiction of an incidence structure),  may look like a [[Graph (mathematics)|graph]], but in a graph an edge has just two endpoints (beyond a vertex a new edge starts), while a line in an incidence structure can be incident to more than two points. In fact, incidence structures are [[hypergraph |hypergraphs]].
 
In an incidence structure there is no concept of a point being between two other points; the order of points on a line is undefined. Compare this with [[ordered geometry]], which does have a notion of betweenness.
 
==Dual structure==
 
If we interchange the role of "points" and "lines" in
 
: ''C'' = (''P'',''L'',''I'')
 
the dual structure
 
: ''C''* = (''L'',''P'',''I''*)
 
is obtained, where ''I''* is the [[inverse relation]] of ''I''. Clearly
 
: ''C''** = ''C''.
 
This is an abstract version of [[Duality (projective geometry)|projective duality]].
 
A structure ''C'' that is [[isomorphic]] to its [[dual]] ''C''* is called ''self-dual''.
 
== Correspondence with hypergraphs ==
[[File:Fano plane with nimber labels.svg|thumb|right|Seven points are elements of seven lines in the [[Fano plane]]]]
Each [[hypergraph]] or [[set system]] can be regarded as an incidence
structure in which the [[universal set]] plays the role of "points", the corresponding [[family of sets]] plays the role of "lines" and the incidence relation is [[Element (mathematics)|set membership]] "∈". Conversely, every incidence structure can be viewed as a hypergraph.
 
===Example: Fano plane===
In particular, let
:''P''&nbsp;=&nbsp;{1,&nbsp;2,&nbsp;3,&nbsp;4,&nbsp;5,&nbsp;6,&nbsp;7},
:''L''&nbsp;=&nbsp;{&nbsp;{1,2,3},&nbsp;{1,4,5},&nbsp;{1,6,7},&nbsp;{2,4,6},&nbsp;{2,5,7},&nbsp;{3,4,7},&nbsp;{3,5,6}&nbsp;}.
 
The corresponding incidence structure is called the [[Fano plane]].
 
The lines are precisely the subsets of the points that consist of three points whose labels add up to zero using [[nim addition]].
 
==Geometric representation==
Incidence structures can be modelled by points and curves in the [[Euclidean plane]] with usual geometric incidence. Some incidence structures admit representation by points and lines. The Fano plane is not one of them since it needs at least one curve.
 
== Levi graph of an incidence structure ==
[[Image:Fano plane-Levi graph.svg|thumb|[[Heawood graph]] with labeling]]
Each incidence structure ''C'' corresponds to a [[bipartite graph]] called the [[Levi graph]] or [[incidence graph]] of the structure. As any bipartite graph is two colorable, the Levi graph can be given a black and white [[graph coloring|vertex coloring]], where black vertices correspond to points and white vertices correspond to lines of ''C''. The edges of this graph correspond to the flags (incident point/line pairs) of the incidence structure.
 
===Example: Heawood graph ===
The Levi graph of the [[Fano plane]] is the [[Heawood graph]]. Since the Heawood graph is [[Connected graph|connected]] and [[vertex-transitive]], there exists an automorphism (such as the one defined by a reflection about the vertical axis in the figure of the Heawood graph) interchanging black and white vertices. This, in turn, implies that the [[Fano plane]] is self-dual.
 
==See also==
*[[Finite geometry]]
*[[Binary relation]]
*[[Combinatorial design]]
*[[Incidence matrix]]
*[[Incidence (geometry)]]
*[[Pappus configuration]]
*[[Projective configuration]]
 
==References==
{{refbegin|2}}
* CRC Press (2000). ''Handbook of discrete and combinatorial mathematics'', (Chapter 12.2), ISBN 0-8493-0149-1
* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', Appendix V: Incidence Structures and Parallelisms, pp 507&ndash;12, [[Marcel Dekker]] ISBN 0-8247-0609-9 .
 
 
{{refend}}
[[Category:Set families]]
[[Category:Combinatorics]]
[[Category:Geometry]]
[[Category:Incidence geometry]]

Latest revision as of 11:45, 28 October 2014

I don't normally look at myself. I appreciate sharing interesting articles with others. I also wish to explore the net for the reason that I believe there is much to check out on the market. When I'm within the web.
|
Name: Junko Vickery
My age: 18
Country: Canada
Town: Beaucanton
Postal code: J0z 1h0
Address: 2303 Rue Des Eglises Est
|
My name is Junko Vickery. I life in Beaucanton (Canada).
|
My name is Junko from Beaucanton doing my final year engineering in Asian Studies. I did my schooling, secured 73% and hope to find someone with same interests in Crocheting.
|
I like Crocheting. Appears boring? Not at all!
I also to learn Bengali in my free time.
|
My name is Junko (31 years old) and my hobbies are Auto audiophilia and Creative writing.
|
Hello! I am Junko. I smile that I can unite to the entire world. I live in Canada, in the QC region. I dream to visit the various countries, to obtain acquainted with intriguing people.
|
I'm Junko (21) from Beaucanton, Canada.
I'm learning Bengali literature at a local university and I'm just about to graduate.
I have a part time job in a post office.
|
Hello!
My name is Junko and I'm a 22 years old boy from Canada.
|
Hi, everybody! My name is Junko.
It is a little about myself: I live in Canada, my city of Beaucanton.
It's called often Northern or cultural capital of QC. I've married 4 years ago.
I have two children - a son (Ernestine) and the daughter (Kristal). We all like Crocheting.
|
Hello!
I'm Bengali female ;=).
I really like Crocheting!
|
Hello from Canada. I'm glad to came across you. My first name is Junko.
I live ipad repair in elk grove village a small town called Beaucanton in east Canada.
I was also born in Beaucanton 26 years ago. Married in March year 2000. I'm working at the post office.
|
My name is Junko Vickery but everybody calls me Junko. I'm from Canada. I'm studying at the college (final year) and I play the Clarinet for 6 years. Usually I choose music from the famous films ;).
I have two sister. I love College football, watching movies and Tai Chi.
|
I'm a 30 years old, married and study at the university (Asian Studies).
In my spare time I try to teach myself Bengali. I have been there and look forward to go there anytime soon. I love to read, preferably on my ebook reader. I like to watch Supernatural and The Vampire Diaries as well as documentaries about anything geological. I like Crocheting.
|
I'm Junko and I live in Beaucanton.
I'm interested in Asian Studies, Crocheting and Bengali art. I like travelling and watching Doctor Who.
|
I'm Junko and I live with my husband and our 2 children ipad repair in hoffman estates Beaucanton, in the QC south area. My hobbies are Microscopy, Knapping and Auto racing.
|
Hello, I'm Junko, a 17 year old from Beaucanton, Canada.
My hobbies include (but are not limited to) Board sports, Videophilia (Home theater) and watching Doctor Who.
|
Hi there! :) My name is Junko, I'm a student studying Asian Studies from Beaucanton, Canada.
|
I'm Junko and I live in a seaside city in northern Canada, Beaucanton. I'm 22 and I'm will soon finish my study at Asian Studies.
|
I am Junko from Beaucanton. I love to play Clarinet. Other hobbies are Crocheting.
|
I am Junko and was born on 27 March 1986. My hobbies are Stone collecting and Coin collecting.
|
My name is Junko and I am studying Social Service and Architecture, Art, and Planning at Beaucanton / Canada.
}

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