Author ORCID Identifier
Year of Publication
Doctor of Philosophy (PhD)
Arts and Sciences
Dr. Margaret A. Readdy
We study a partial ordering on pairings called the uncrossing poset, which first appeared in the literature in connection with a certain stratified space of planar electrical networks. We begin by examining some of the relationships between the uncrossing poset and Catalan combinatorics, and then proceed to study the structure of lower intervals. We characterize the lower intervals in the uncrossing poset that are isomorphic to the face lattice of a cube. Moving up in complexity certain lower intervals are isomorphic to the poset of simple vertex labeled minors of an associated graph.
Inspired by this structure, we define a notion of minors for lattices enriched with a generating set that abstracts the notion of simple vertex labeled minors of a graph. We can associate a generator-enriched lattice to any polymatroid, a far reaching generalization of graphs, and we show that conversely, any generator-enriched lattice has an associated polymatroid. The generator-enriched lattice encodes the simple information of the closure operator of the polymatroid analagous to how a geometric lattice encodes the simple information of a matroid. For a generator-enriched lattice associated to a graph we show the minors of the generator-enriched lattice are in bijection with the simple vertex labeled minors of the graph. This bijection is generalized to any generator-enriched lattice and its associated polymatroid.
We proceed to study a partial order structure on the minors of a given generator-enriched lattice called the minor poset whose relations correspond to performing deletions and contractions. A construction for minor posets in terms of the zipping operation introduced by Reading is given. This construction implies any minor poset is isomorphic to the face poset of a regular CW sphere, and in particular, implies minor posets are Eulerian. This construction also yields cd-index inequalities for minor posets. We characterize the generator-enriched lattices whose minor poset is itself a lattice as meet-distributive lattices avoiding a single forbidden minor. As a special case, minor posets of distributive lattices avoiding this forbidden minor are isomorphic to the face lattice of the order polytope of the dual of the poset of join-irreducibles.\
The deletion and contraction operations of generator-enriched lattices do not commute. We introduce a modified version of contractions, called weak contractions that does commute with deletions and from this operation define weak minor posets whose relations correspond to performing deletions and weak contractions. The theory of weak minor posets closely parallels that of minor posets. Most notably, weak minor posets are shown to be complemented lattices and strong maps between generator-enriched lattices induce meet-preserving maps between the weak minor posets in analogy with the zipping construction for minor posets. We characterize graded weak minor posets and induce EL-labelings for weak minor posets of generator-enriched lattices whose minors each have an EL-labeling. In particular, we find any graded weak minor poset is also shellable.
Digital Object Identifier (DOI)
This research was supported by the Clifford J. Swauger, Jr. Graduate Fellowship in 2020.
Gustafson, William, "Lattice minors and Eulerian posets" (2023). Theses and Dissertations--Mathematics. 96.