Decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata

TL;DR

This paper proves that it is decidable whether a finite semigroup or automaton has Krohn-Rhodes complexity 1, resolving a long-standing open problem by establishing sharp lower bounds using advanced algebraic methods.

Contribution

It establishes the decidability of Krohn-Rhodes complexity 1 for finite semigroups and automata, advancing understanding of their structural decomposition.

Findings

Proves decidability of Krohn-Rhodes complexity c=1 for finite semigroups and automata.

Shows the lower bounds in previous work are sharp using profinite methods.

Utilizes results from McCammond (1991, 2001) to achieve main result.

Abstract

When decomposing a finite semigroup into a wreath product of groups and aperiodic semigroups, complexity measures the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for almost 60 years. The main result of this paper proves decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata. This is achieved by showing the lower bounds in work by Henckell, Rhodes and Steinberg from 2012 is sharp using profinite methods and results of McCammond from 1991 and 2001.