Characterizations ot partition lattices
John F. Randolph, Kenneth P. Bogart · 2005
In this paper we present a structural characterization of partition lattices: If L has a modular copoint and if [p, 1] ~ H, 1 for all points p c L, then L ~ Hn for n --> 5. From this we observe that if a super-solvable geometric lattice L and all of its upper intervals have the characteristic polynomial of a partition lattice, then L is isomorphic to a partition lattice. Although the first result is known [1], the proof presented here is new. We shall use W k ( L ) (called the Whitney numbers of the second kind) to denote the number of elements in a geometric lattice L which have rank k and S(n, k ) = W,-k. The words point, line, copoint and coline refer to elements of rank 1, 2, n 1 and n 2 in a geometric lattice of rank n, while atom and coatom refer to elements in quotient lattices.