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Matroid theory borrows extensively from the terminology of [[linear algebra]] and [[graph theory]], largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, [[topology]], [[combinatorial optimization]], [[network theory]] and [[coding theory]].<ref name=Neel2009>{{cite journal|last1=Neel|first1=David L.|last2=Neudauer|first2=Nancy Ann|title=Matroids you have known|journal=Mathematics Magazine|date=2009|volume=82|issue=1|pages=26-41|url=http://www.maa.org/sites/default/files/pdf/shortcourse/2011/matroidsknown.pdf|accessdate=4 October 2014}}</ref><ref name=Kashyap2009>{{cite web|last1=Kashyap|first1=Navin|last2=Soljanin|first2=Emina|last3=Vontobel|first3=Pascal|title=Applications of Matroid Theory and Combinatorial Optimization to Information and Coding Theory|url=https://www.birs.ca/workshops/2009/09w5103/report09w5103.pdf|website=www.birs.ca|accessdate=4 October 2014}}</ref>
===Finite geometries===
{{main|Finite geometry}}
A [[finite geometry]] is any [[geometry|geometric]] system that has only a [[finite set|finite]] number of [[point (geometry)|points]].
The familiar [[Euclidean geometry]] is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the [[pixel]]s are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite [[projective space|projective]] and [[affine space]]s because of their regularity and simplicity. Other significant types of finite geometry are finite [[Möbius plane|Möbius or inversive plane]]s and [[Laguerre plane]]s, which are examples of a general type called [[Benz plane]]s, and their higher-dimensional analogs such as higher finite [[inversive geometry|inversive geometr]]ies.
Finite geometries may be constructed via [[linear algebra]], starting from [[vector space]]s over a [[finite field]]; the affine and [[projective plane]]s so constructed are called [[Galois geometry|Galois geometries]]. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite [[projective space]] of dimension three or greater is [[isomorphism|isomorphic]] to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the [[non-Desarguesian plane]]s. Similar results hold for other kinds of finite geometries.
== See also ==
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== References ==
{{reflist}}
==Further reading==
* {{ cite book | last1=Bannai | first1=Eiichi | <!-- authorlink1=Eiichi Bannai | authorlink2= Tatsuro Ito --> | last2=Ito | first2=Tatsuro | title=Algebraic combinatorics I: Association schemes | publisher=The Benjamin/Cummings Publishing Co., Inc. | ___location=Menlo Park, CA | year=1984 | pages=xxiv+425 | isbn=0-8053-0490-8 | mr=0882540 }}
* {{cite book|first=C. D.| last=Godsil|authorlink = Chris Godsil|title=Algebraic Combinatorics|publisher=Chapman and Hall|year=1993|___location=New York|ISBN=0-412-04131-6 | mr=1220704 }}
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[[Category:Algebraic combinatorics|*]]
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