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Finite Presentability of Brin-Higman-Thompson Monoids via Free Jónsson-Tarski Algebras

Bill de Witt, Luna Elliott

TL;DR

The paper resolves the finite presentability question for the Brin–Higman–Thompson monoids $totM_{k,r}$ by realizing them as Endomorphism monoids of free multidimensional Jónsson‑Tarski algebras $\mathbb{F}_{n,k,r}$ and interpreting elements as rewrite rules. It constructs a natural isomorphism between $\operatorname{tot} nM_{k,r}$ and $\operatorname{End}(\mathbb{F}_{n,k,r})$ via a tree‑based embedding, and then develops a rewrite‑theoretic framework with labelled generating sets to obtain a finite presentation. The authors provide a detailed finite‑presentation proof, including an explicit presentation for the classical case $totM_{2,1}$ with a lengthy relator list. The approach connects algebraic endomorphisms with combinatorial structures (prefix codes, pseudotrees) and yields a concrete method to obtain finite presentations for higher‑dimensional generalizations. These results extend the understanding of finite presentability from the classic one‑dimensional setting to a broad multidimensional family of monoids, with potential implications for computational and structural aspects of these groups and monoids.

Abstract

We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional Jónsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". Using these representations, we show that the monoids are finitely presented.

Finite Presentability of Brin-Higman-Thompson Monoids via Free Jónsson-Tarski Algebras

TL;DR

The paper resolves the finite presentability question for the Brin–Higman–Thompson monoids by realizing them as Endomorphism monoids of free multidimensional Jónsson‑Tarski algebras and interpreting elements as rewrite rules. It constructs a natural isomorphism between and via a tree‑based embedding, and then develops a rewrite‑theoretic framework with labelled generating sets to obtain a finite presentation. The authors provide a detailed finite‑presentation proof, including an explicit presentation for the classical case with a lengthy relator list. The approach connects algebraic endomorphisms with combinatorial structures (prefix codes, pseudotrees) and yields a concrete method to obtain finite presentations for higher‑dimensional generalizations. These results extend the understanding of finite presentability from the classic one‑dimensional setting to a broad multidimensional family of monoids, with potential implications for computational and structural aspects of these groups and monoids.

Abstract

We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional Jónsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". Using these representations, we show that the monoids are finitely presented.
Paper Structure (12 sections, 23 theorems, 74 equations, 1 figure)

This paper contains 12 sections, 23 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions of Objects
  3. Notes on Notation
  4. Words, Trees, and Prefix Codes
  5. Varieties
  6. Free objects in the multidimensional Jónsson-Tarski varieties
  7. Constructing the Isomorphism
  8. The Rewrite Perspective
  9. Finite Presentability
  10. Proving presentability
  11. Explicit presentation for tot$M_{2,1}$
  12. Closing Comments and Questions

Key Result

Theorem 1.1

For $r, n, k \in \mathbb{N}$, with $r, n \geq 1$ and $k \geq 2$, the following monoids are isomorphic:

Figures (1)

  • Figure 1: Pictoral representation of Proposition \ref{['prop:nVk1']} iii)

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Alphabets and $nT_{k,r}$
  • Definition 2.2: Cantor Spaces
  • Definition 2.3: Shrubs, Shrubberies, and Depth
  • Definition 2.4: The Partial Order
  • Definition 2.5: Cones
  • Definition 2.6: Pseudotrees and Complete Prefix Codes
  • Remark 2.7
  • Definition 2.8: Pseudotree of a Prefix Code
  • ...and 57 more