Thomas Hugh - A bijection between NBB sets of atoms and descending chains in admissible lattices

A finite graded lattice is admissible if it has a labelling of its join-irreducibles with certain good properties. This labelling induces an edge-labelling, and it is known that the Mobius function of the lattice can be computed in terms of the numbers of maximal chains in the lattice which are descending relative to the edge-labelling. So far, this is based on papers of Stanley's from the 1970s. Recently, Blass and Sagan introduced the idea of NBB sets of atoms of a lattice, and showed that the Mobius function of a lattice could be computed in terms of them. We construct a bijection between the descending chains and the relevant collection of NBB sets. Also, we define quasi-admissibility of a (not necessarily graded) lattice, a generalization of admissibility, for which the above bijection also holds. We show that any lattice with a maximal left-modular chain is quasi-admissible.