Transversal (combinatorics)

For other uses, see Transversal (combinatorics) (disambiguation).

In combinatorial mathematics, given a collection C of disjoint 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 $G=H\times K$, 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.