Transversal

In combinatorial mathematics, given a collection "C" of sets, a transversal is a set containing exactly one element from each member of the collection: it is a section of the quotient map induced by the collection. If the original sets are not disjoint, there are several different definitions. One variation 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. A less restrictive definition requires that the transversal just has a non-empty intersection with each member of "C".

Examples

As an example of the disjoint-sets meaning of "transversal",in group theory, given a subgroup "H" of a group "G", a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of "H".

Given a direct product of groups $G\; =\; H\; imes\; K$, then "H" is a transversal for the cosets of "K", and conversely.

* The marriage theorem gives necessary and sufficient conditions for possibly overlapping subsets to have a transversal.

References

*Mirsky, Leon (1971). "Transversal Theory: An account of some aspects of combinatorial mathematics." Academic Press. ISBN 0-12-498550-5.