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Complexity of Finite Semigroups: History and Decidability

StuarT Margolis, John Rhodes, Anne Schilling

Abstract

In recent papers, Margolis, Rhodes and Schilling proved that the complexity of a finite semigroup is computable. This solved a problem that had been open for more than 50 years. The purpose of this paper is to survey the basic results of Krohn-Rhodes complexity of finite semigroups and to outline the proof of its computability.

Complexity of Finite Semigroups: History and Decidability

Abstract

In recent papers, Margolis, Rhodes and Schilling proved that the complexity of a finite semigroup is computable. This solved a problem that had been open for more than 50 years. The purpose of this paper is to survey the basic results of Krohn-Rhodes complexity of finite semigroups and to outline the proof of its computability.
Paper Structure (10 sections, 28 theorems, 3 equations)

This paper contains 10 sections, 28 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Upper and Lower Bounds to Complexity
  3. Upper Bounds
  4. Lower Bounds
  5. The Complexity Theory of Inverse Semigroups
  6. Tools Used for the Proof of Computability of Complexity
  7. Transitive Representations of Finite Semigroups
  8. The Presentation Lemma and Flows
  9. Outline of the Proof for Computability of Complexity 1
  10. Outline of the Proof for Computability of Arbitrary Complexity

Key Result

Theorem 1.2

Every finite semigroup $S$ divides a wreath product of groups and aperiodic semigroups. One can choose the groups to be simple groups that divide $S$. We can choose the aperiodic semigroups to be the flip-flop $\mathsf{RZ}(2)^1$. Furthermore, $S$ is prime if and only if either $S$ is a simple group

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2: Krohn--Rhodes 1962
  • Corollary 1.3
  • Definition 1.4
  • Theorem 2.1: The Depth Decomposition Theorem
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 34 more