Profinite tree sets

TL;DR

The paper develops a comprehensive theory of profinite tree sets, showing that infinite tree sets arise as inverse limits of finite tree sets and that every profinite tree set is itself the inverse limit of such finite quotients. It introduces a finite-quotient construction $ au/D$ via branch-closed selections $D$ and proves that, under chain-completeness and splittability, these quotients form a directed system whose inverse limit recovers the original profinite tree set. A precise combinatorial characterization of profinite tree sets is obtained (chain-complete, splittable, no infinite regular splitting star, and finite $C(s,s')$ for regular $s,s'$), and the representation of profinite trees by bipartitions of orientations is extended to the profinite setting, including cases with splitting stars. These results extend the tree-of-tangles framework from finite to profinite separation systems and provide tools for representing and manipulating profinite tree sets in combinatorial and representation-theoretic terms.

Abstract

Tree sets are posets with additional structure that generalize tree-like objects in graphs, matroids, or other combinatorial structures. They are a special class of abstract separation systems. We study infinite tree sets and how they relate to the finite tree sets they induce, and obtain a characterization of infinite tree sets in combinatorial terms.

Theorems & Definitions (82)

• Theorem 1.1
• Lemma 2.1: Extension Lemma
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 3.1: ProfiniteASS