Representations of infinite tree-sets

TL;DR

This paper extends the representation theory of finite tree sets to infinite ones, characterizing which can be represented by infinite trees and introducing a topological generalization to encompass all infinite tree sets.

Contribution

It provides a characterization of infinite tree sets representable by infinite trees and introduces a topological framework for representing all infinite tree sets.

Findings

Characterization of tree sets representable by infinite trees.

Introduction of a topological generalization for infinite trees.

Every infinite tree set can be represented by a suitable topological tree-like space.

Abstract

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ${\omega+1}$. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.