Steiner Complexes, Matroid Ports, and Shellability

We study the theory of matroid Steiner families, which were introduced in a recent paper of Colbourri and Pulleyblank as abstractions of Steiner trees on graphs. Our primary result is that these families are equivalent to matroid ports.Using this equivalence, we relate these Steiner families to two fundamental matroid-theoretic constructions namely, single-element extensions and elementary quotients. We also discuss a new and very general class of Steiner families, which arise as minimal supports of affine systems. The latter half of the paper deals with the structure of Steiner complexes, which are independence systems associated with these Steiner families. We develop the notion of an S-partition of a simplicial complex which extends the notion of a shelling of a pure simplicial complex. We develop an S-partitioning theory for Steiner complexes generalizing the well-known shellability theory for the collection of independent sets of a matroid.