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The Expansion Problem for Infinite Trees

Achim Blumensath

TL;DR

This work investigates the Expansion Problem for tree algebras over infinite trees, seeking to extend partial, tree-defined products to universal ones. It develops two complementary tools—evaluations and consistent labellings—to analyze when such expansions exist and are unique, with MSO-definable algebras serving as the main testing ground. The authors establish a complete solution for thin trees, showing density and unique expansions for $\mathbb T^{\mathrm{wilke}} \subseteq \mathbb T^{\mathrm{thin}}$, and obtain partial results for non-thin cases such as $\mathbb T^{\mathrm{reg}} \subseteq \mathbb T$, while outlining key open problems and promising directions (e.g., generalising Simon’s Factorisation Tree Theorem to trees and developing Green’s relations for tree algebras). The work connects Ramsey-type tree combinatorics, automata theory, and algebraic language theory to advance understanding of when algebraic expansions exist and are uniquely determined for infinite trees.

Abstract

We study Ramsey like theorems for infinite trees and similar combinatorial tools. As an application we consider the expansion problem for tree algebras.

The Expansion Problem for Infinite Trees

TL;DR

This work investigates the Expansion Problem for tree algebras over infinite trees, seeking to extend partial, tree-defined products to universal ones. It develops two complementary tools—evaluations and consistent labellings—to analyze when such expansions exist and are unique, with MSO-definable algebras serving as the main testing ground. The authors establish a complete solution for thin trees, showing density and unique expansions for , and obtain partial results for non-thin cases such as , while outlining key open problems and promising directions (e.g., generalising Simon’s Factorisation Tree Theorem to trees and developing Green’s relations for tree algebras). The work connects Ramsey-type tree combinatorics, automata theory, and algebraic language theory to advance understanding of when algebraic expansions exist and are uniquely determined for infinite trees.

Abstract

We study Ramsey like theorems for infinite trees and similar combinatorial tools. As an application we consider the expansion problem for tree algebras.
Paper Structure (12 sections, 51 theorems, 235 equations, 5 figures)

This paper contains 12 sections, 51 theorems, 235 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Tree algebras
  3. Partition theorems for trees
  4. Expansions and dense submonads
  5. Expansion Problem.
  6. Evaluations
  7. Thin trees
  8. Evaluations with merging
  9. Consistent labellings
  10. Unambiguous algebras
  11. Branch-continuous algebras
  12. Conclusion

Key Result

Theorem 7

Let $\mathbb T^0 \subseteq \mathbb T^\times$.

Figures (5)

  • Figure 1: The flattening operation: $t$ and $\mathrm{flat}(t)$
  • Figure 2: A function $\sigma$ (along a single branch of $t$) and the corresponding relation $\sqsubseteq_\sigma$, indicated via the grey bars.
  • Figure 3: The trees $t$ and $t'$.
  • Figure 4: An evaluation (on the right) encoded by a function (on the left). For simplicity, we have omitted the labels from $A$ and drawn just the $\tau$-labelling.
  • Figure 5: (a) Left: the plays $\nu$ and $\nu'$. Dashed lines represent plays in $\mathcal{G}'$, solid ones plays in $\mathcal{G}$. (b) Right: the plays $\rho_i$ and $\nu'_i$, projected to the game $\mathcal{G}$.

Theorems & Definitions (142)

  • Definition 1
  • Definition 2
  • Remark
  • Definition 3
  • Remark
  • Definition 4
  • Remark
  • Definition 5
  • Example 1
  • Definition 6
  • ...and 132 more