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Some properties of Higman-Thompson monoids and digital circuits

J. C. Birget

TL;DR

The paper builds a unified framework tying Higman–Thompson monoids to acyclic circuit models, showing that monoids such as $M_{2,1}$ and ${\sf tot}M_{2,1}$ naturally arise as right-ideal morphisms and as circuit-input/output maps. It proves finite generation for ${\cal RM}^{\sf fin}_2$ and related monoids, while establishing non-finite generation for several fixed-length and plep/tlep variants. The work also develops completion procedures (deterministic, inverse-homomorphic, generator-based, and circuit-based), and provides a robust representation theory: any element can be decomposed into a finite unambiguous union of partial circuits with disjoint domains, bridging algebraic and circuit viewpoints. Collectively, these results reveal a deep equivalence between Thompson monoids and circuit-model computations, with concrete algorithms for completion and decomposition and a detailed complexity landscape for related decision problems.

Abstract

We define various monoid versions of the R. Thompson group $V$, and prove connections with monoids of acyclic digital circuits. We show that the monoid $M_{2,1}$ (based on partial functions) is not embeddable into Thompson's monoid ${\sf tot}M_{2,1}$, but that ${\sf tot}M_{2,1}$ has a submonoid that maps homomorphically onto $M_{2,1}$. This leads to an efficient completion algorithm for partial functions and partial circuits. We show that the union of partial circuits with disjoint domains is an element of $M_{2,1}$, and conversely, every element of $M_{2,1}$ can be decomposed efficiently into a union of partial circuits with disjoint domains.

Some properties of Higman-Thompson monoids and digital circuits

TL;DR

The paper builds a unified framework tying Higman–Thompson monoids to acyclic circuit models, showing that monoids such as and naturally arise as right-ideal morphisms and as circuit-input/output maps. It proves finite generation for and related monoids, while establishing non-finite generation for several fixed-length and plep/tlep variants. The work also develops completion procedures (deterministic, inverse-homomorphic, generator-based, and circuit-based), and provides a robust representation theory: any element can be decomposed into a finite unambiguous union of partial circuits with disjoint domains, bridging algebraic and circuit viewpoints. Collectively, these results reveal a deep equivalence between Thompson monoids and circuit-model computations, with concrete algorithms for completion and decomposition and a detailed complexity landscape for related decision problems.

Abstract

We define various monoid versions of the R. Thompson group , and prove connections with monoids of acyclic digital circuits. We show that the monoid (based on partial functions) is not embeddable into Thompson's monoid , but that has a submonoid that maps homomorphically onto . This leads to an efficient completion algorithm for partial functions and partial circuits. We show that the union of partial circuits with disjoint domains is an element of , and conversely, every element of can be decomposed efficiently into a union of partial circuits with disjoint domains.
Paper Structure (28 sections, 58 theorems, 8 equations)

This paper contains 28 sections, 58 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminary monoids
  3. Circuits and monoids
  4. Circuits
  5. Partial circuits
  6. Various definitions of various Thompson monoids
  7. Equivalence relations between prefix codes
  8. Congruence relations on ${\cal RM}^{\sf fin}$
  9. Defining Thompson monoids
  10. Finite generation of ${\cal RM}^{\sf fin}_2$
  11. Normal functions
  12. Proof that ${\cal RM}^{\sf fin}_2$ is finitely generated
  13. ${\sf tlep}{\cal RM}^{\sf fin}_2$, $\,{\sf tlep}M_{2,1}$, $\,{\sf plep}{\cal RM}^{\sf fin}_2$, and $\,{\sf plep}M_{2,1}$, are not finitely generated
  14. Some algebraic properties of the monoids $M_{2,1}$, $\,{\sf tot}M_{2,1}$, $\,{\sf sur}M_{2,1}$, and $\,{\sf plep}M_{k,1}$
  15. Regularity
  16. ...and 13 more sections

Key Result

Lemma 3.4

(equivalent definition of input-length). Let $\Gamma$ be a finite set such that $\,\Gamma \cup \tau$ generates ${\cal RM}^{\sf fin}$ (or ${\sf tot}{\cal RM}^{\sf fin}$, ${\sf tfl}{\cal RM}^{\sf fin}$, ${\sf pfl}{\cal RM}^{\sf fin}$), and let $\,u = u_k \,\ldots\,u_1 \in$$(\Gamma \cup \tau)^*$ with $

Theorems & Definitions (80)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 70 more