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The Todd-Coxeter Algorithm for Semigroups and Monoids

T. D. H. Coleman, J. D. Mitchell, F. L. Smith, M. Tsalakou

Abstract

In this paper we provide an account of the Todd-Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd-Coxeter algorithm for groups.

The Todd-Coxeter Algorithm for Semigroups and Monoids

Abstract

In this paper we provide an account of the Todd-Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd-Coxeter algorithm for groups.
Paper Structure (24 sections, 22 theorems, 23 equations, 16 figures, 17 tables)

This paper contains 24 sections, 22 theorems, 23 equations, 16 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Congruence enumeration
  4. Word graphs
  5. The definition
  6. Validity
  7. Completeness, determinism, and compatibility
  8. The proofs of \ref{['validity']} and \ref{['cor-tc-terminates']}
  9. Monoids not defined by a presentation
  10. The HLT strategy
  11. The Felsch strategy
  12. First version
  13. Second version
  14. Implementation issues
  15. Further variants
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $S$ be a semigroup, let $\rho$ be a congruence of $S$, and let $\sigma$ be a right congruence of $S$ such that $\rho \subseteq \sigma$. The right actions of $S$ on $(S / \rho) / (\sigma / \rho)$ and $S / \sigma$ defined by are isomorphic.

Figures (16)

  • Figure 2.1: A commutative diagram illustrating a homomorphism of actions where $X\times S \longrightarrow Y\times S$ is the function defined by $(x, s) \mapsto ((x)\lambda, s)$.
  • Figure 4.1: The commutative diagram in the proof of \ref{['validity']}(a).
  • Figure 5.1: The right Cayley graph of the monoid $M$ from \ref{['ex-cayley-digraph']} with respect to the generating set $\{a, b, c\}$; see \ref{['table-ex-cayley-digraph']} for a representative word for each node. Purple arrows correspond to edges labelled $a$, gray labelled $b$, and pink labelled $c$.
  • Figure 5.2: The output $(\Gamma_{i}, \kappa_{i})$ of each step in \ref{['ex-cayley-digraph-cong']}. Purple arrows correspond to $a$, gray to $b$, pink to $c$, shaded nodes of the same colour belong to $\kappa_{i}$, and unshaded nodes belong to singleton classes; see \ref{['table-ex-cayley-digraph']} for a representative word for each node. A dashed edge with a single arrowhead denotes an edge that is obtained from TC2 or TC3, solid edges correspond to edges that existed at the previous step.
  • Figure 6.1: The output $(\Gamma_{i}, \kappa_{i})$ of each step in the HLT strategy in \ref{['ex-tc-1']}. Purple arrows correspond to $a$, gray to $b$, pink to $c$, shaded nodes of the same colour belong to $\kappa_{i}$, and unshaded nodes belong to singleton classes; see \ref{['table-ex-tc-1b']} for a representative word for each node. A dashed edge with a double arrowhead indicates the edge being defined in TC1, a dashed edge with a single arrowhead denotes an edge that is obtained from TC2 or TC3, solid edges correspond to edges that existed at the previous step.
  • ...and 11 more figures

Theorems & Definitions (45)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Theorem 4.1
  • Corollary 4.1
  • Lemma 4.2
  • Proposition 4.3
  • Lemma 4.4
  • ...and 35 more