Supersolvable Frame-matroid and Graphic-lift Lattices

A geometric lattice is a frame if its matroid, possibly after enlargement, has a basis such that every atom lies under a join of at most two basis elements. Examples include all subsets of a classical root system. Using the fact that nitary frame matroids are the bias matroids of biased graphs, we characterize modular coatoms in frames of nite rank and we describe explicitly the frames that are supersolvable. We apply the characterizations to three kinds of example: one generalizes the root system D n and near-Dowling lattices ; one has for characteristic polynomials the enumerators of separated circular partial permutations; and one is the family of bicircular matroids. A geometric lattice is a graphic lift if it can be extended to contain an atom whose upper interval is graphic. We characterize modular coatoms in and supersolvability of graphic lifts of nite rank and we examine families analogous to the frame examples.