Extensions and presentations of transversal matroids

A transversal matroid M can be represented by a collection of sets, called a presentation of M, whose partial transversals are the independent sets of M. Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (single-element) transversal extensions of M and extensions of presentations of M. We show that a presentation of M is minimal if and only if different extensions of it give different extensions of M; also, all transversal extensions of M can be obtained by extending the minimal presentations of M. We also begin to explore the partial order that the weak order gives on the transversal extensions of M.