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Decidability of Krohn-Rhodes complexity for all finite semigroups and automata

Stuart Margolis, John Rhodes, Anne Schilling

Abstract

The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for more than 50 years. The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0. This builds on our previous work for complexity 1. In that paper we proved using profinite methods and results on free Burnside semigroups by McCammond and others that the lower bound from a 2012 paper by Henckell, Rhodes and Steinberg is precise for complexity 1. In this paper we define an improved version of the lower bound from the 2012 paper and prove that it is exact for arbitrary complexity.

Decidability of Krohn-Rhodes complexity for all finite semigroups and automata

Abstract

The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for more than 50 years. The main result of this paper proves that it is decidable whether a semigroup has complexity k for any k greater than or equal to 0. This builds on our previous work for complexity 1. In that paper we proved using profinite methods and results on free Burnside semigroups by McCammond and others that the lower bound from a 2012 paper by Henckell, Rhodes and Steinberg is precise for complexity 1. In this paper we define an improved version of the lower bound from the 2012 paper and prove that it is exact for arbitrary complexity.

Paper Structure

This paper contains 10 sections, 28 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. History and known results
  3. Preliminaries, main result, and proof outline
  4. Background
  5. The connection between Rhodes lattices and set-partition lattices
  6. The monoid of closure operators on a lattice
  7. Flows and the Presentation Lemma Flow Decomposition Theorem
  8. Flow monoids and evaluation transformation semigroups
  9. Decidability of complexity for all finite semigroups and automata
  10. Proof of decidability

Key Result

Theorem 1.1

Let $S$ be a $\operatorname{GM}$ semigroup.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2: The Presentation Lemma Flow Decomposition Theorem
  • Theorem 1.3: Margolis, Rhodes, Schilling 2023
  • Theorem 1.4: Margolis, Rhodes, Schilling 2023
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 33 more