An Affine Representation for Transversal Geometries

Pregeometries (matroids) whose independent sets are the partial matchings of a relation (transversal pregeometries) can be canonically imbedded in a free-simplicial pregeometry (one whose points lie freely on flats spanned by a simplex). Conversely, all subgeometries of such free-simplicial pregeometries are transversal. Free-simplicial pregeometries are counted and their duals are naturally constructed and shown to be free-simplicial (showing self-dual free-simplexes corrspond to quasisymmetric relations). For more general transversal pregeometries, modular flats are characterized and transversal contractions are exemplified. Binary transversal pregeometries and their contractions (the class of binary gammoids) are shown to be the class of series-parallel networks, providing insight for further characterizations of (coordinatized) gammoids by excluded minors. Theorem. All principal transversal pregeometries and their truncations have critical exponent at most 2.