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

TL;DR

This paper proves that it is decidable whether a finite semigroup has a given Krohn-Rhodes complexity, resolving a long-standing open problem and extending previous results for complexity 1 to all complexities.

Contribution

It establishes the decidability of Krohn-Rhodes complexity for all finite semigroups, providing an effective method to determine the minimal number of groups needed.

Findings

Decidability of Krohn-Rhodes complexity for any k.

An improved lower bound that is exact for all complexities.

Extension of previous complexity 1 results to arbitrary complexity.

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.