Oriented Hypergraphic Matrix-tree Type Theorems and Bidirected Minors via Boolean Order Ideals

Given an integer matrix there is a natural oriented hypergraph associated to it. We examine restrictions of incidence-preserving maps to produce Sachs-Chaiken type All Minors Matrix-tree Theorems for oriented hypergraphic Laplacian and adjacency matrices. When the incidence structure is restricted to bidirected graphs the minor calculations are shown to correspond to principal order ideals of signed boolean lattices, and classical results are generalized or reclaimed.