Algebraic combinatorics: Difference between revisions

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A [[matroid]] is a structure that captures and generalizes the notion of [[linear independence]] in [[vector space]]s. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.

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|author2-link= Nancy Neudauer |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|accessdateaccess-date=4 October 2014|doi=10.4169/193009809x469020}}</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|accessdateaccess-date=4 October 2014}}</ref>

===Finite geometries===

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==Further reading==

*{{ cite book | last1=Bannai | first1=Eiichi | 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|ref=harv }}

* {{cite book|editor-first1=Louis J. |editor-last1=Billera|editor1-link=Louis Billera|editor-first2= Anders|editor-last2= Björner|editor2-link=Anders Björner|editor-first3= Curtis|editor-last3= Greene|editor3-link=Curtis Greene | editor-first4= Rodica|editor-last4= Simion|editor4-link=Rodica Simion|editor-first5=Richard P.|editor-last5= Stanley | editor5-link=Richard P. Stanley |url=http://library.msri.org/books/Book38/index.html|title=New Perspectives in Algebraic Combinatorics|series= MSRI Publications|volume= 38|publisher= [[Cambridge University Press]]|year= 1999}}

*{{cite book|first=Chris D.| last=Godsil|author-link = Chris Godsil|title=Algebraic Combinatorics|publisher=Chapman and Hall|year=1993|___location=New York|ISBN=0-412-04131-6 | mr=1220704|ref=harv }}

[[Cambridge University Press]]|year= 1999}}

*{{cite book|first=Chris D.| last=Godsil|author-link = Chris Godsil|title=Algebraic Combinatorics|publisher=Chapman and Hall|year=1993|___location=New York|ISBN=0-412-04131-6 | mr=1220704|ref=harv }}

* Takayuki Hibi, ''Algebraic combinatorics on convex polytopes'', Carslaw Publications, Glebe, Australia, 1992

*[[Melvin Hochster]], ''Cohen-Macaulay rings, combinatorics, and simplicial complexes''. Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp.&nbsp;171–223. Lecture Notes in Pure and Appl. Math., vol. 26, Dekker, New York, 1977.