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{{Other uses|Transversal (disambiguation)}}
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In [[combinatorics|combinatorial]] mathematics, given a collection ''C'' of [[set theory|sets]], a '''transversal''' is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of ''C'' (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal. One variation, the one that mimics the situation when the sets are mutually disjoint, is that there is a [[bijection]] ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a '''system of distinct representatives'''. The other, less commonly used, possibility does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. Loosely speaking, in this situation the members of the ''system of representatives'' are not necessarily distinct.<ref>{{harvnb|Roberts|Tesman|2009|loc=pg. 692}}</ref><ref>{{harvnb|Brualdi|2010|loc=pg. 322}}</ref>
 
A '''partial transversal''' is a set containing at most one element from each member of the collection, or (in the stricter form of the concept) a set with an injection from the set to ''C''.
 
The transversals of a finite collection ''C'' of finite sets form the basis sets of a [[matroid]], the "transversal matroid" of ''C''. The independent sets of the transversal matroid are the partial transversals of ''C''.<ref>{{citation|page=48|title=Matroid Theory|volume=3|series=Oxford graduate texts in mathematics|first=James G.|last=Oxley | authorlink = James Oxley|publisher=Oxford University Press|year=2006|isbn=978-0-19-920250-8}}.</ref>
 
A generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of ''C''. An example of this would be a [[Bernstein set]], which is defined as a set that has a non-empty intersection with each set of ''C'', but contains no set of ''C'', where ''C'' is the collection of all [[perfect sets]] of a topological [[Polish space]].
 
==Examples==
In [[group theory]], given a [[subgroup]] ''H'' of a group ''G'', a right (respectively left) transversal is a [[Set (mathematics)|set]] containing exactly one element from each right (respectively left) [[coset]] of ''H''. In this case, the "sets" (cosets) are mutually disjoint.
 
Given a [[direct product of groups]] <math>G = H \times K</math>, then ''H'' is a transversal for the cosets of ''K''.
 
* [[Hall's marriage theorem]] gives necessary and sufficient conditions for a finite collection of not necessarily distinct, but non-empty sets to have a transversal.
 
==Category theory==
In the language of [[category theory]], a '''transversal''' of a collection of mutually disjoint sets is a [[Section (category theory)|section]] of the [[quotient map]] induced by the collection.
 
==See also==
*[[Axiom of choice]]
*[[Blocking set]]
*[[Permanent]]
 
==Notes==
{{reflist}}
 
==References==
* {{citation|last=Brualdi|first=Richard A.|title=Introductory Combinatorics|edition=5th|year=2010|publisher=Prentice Hall|place=Upper Saddle River, NJ |isbn=0-13-602040-2}}
*[[Eugene Lawler|Lawler, E. L.]] Combinatorial Optimization: Networks and Matroids. 1976.
*[[Leon Mirsky|Mirsky, Leon]] (1971). ''Transversal Theory: An account of some aspects of combinatorial mathematics.'' Academic Press. ISBN 0-12-498550-5.
* {{citation|last1=Roberts|first1=Fred S.|last2=Tesman|first2=Barry|title=Applied Combinatorics|edition=2nd|publisher=CRC Press|place=Boca Raton|year=2009|isbn=978-1-4200-9982-9}}
 
[[Category:Combinatorics]]
[[Category:Group theory]]
 
{{Combin-stub}}

Latest revision as of 05:25, 30 November 2014

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