Perfect orderings on Bratteli diagrams II: general Bratteli diagrams

Abstract

We continue our study of orderings on Bratteli diagrams started in previous work, joint with Jan Kwiatkowski, where Bratteli diagrams of finite rank were considered. We extend the notions of languages, permutations (called correspondences in this paper), skeletons and associated graphs to the case of general Bratteli diagrams, and show their relevance to the study of perfect orderings: those that support Vershik maps; in particular, perfect orderings with several extremal paths. A perfect ordering comes equipped with a skeleton and a correspondence, and conversely, given a skeleton and correspondence, we describe explicitly how to construct perfect orderings, by showing that paths in the associated directed graphs determine the language of the order. We describe an explicit algorithmic method to create perfect orderings on Bratteli diagrams based on the study of certain relations between the entries of the diagram's incidence matrices and properties of the associated graphs, with the latter relations characterizing diagrams which support perfect orderings. Also, we apply the notions of skeletons and associated graphs, to give a new combinatorial proof of the fact that diagrams supporting perfect orderings with k maximal paths have a direct sum of k-1 copies of the integers contained in their infinitesimal subgroup. Under certain conditions, we show that a similar result holds if the diagram supports countably many maximal paths. Our results are illustrated by numerous examples.