2024/06/26 by Stuart Margolis, Margolis, Stuart, John Rhodes +3 · 2 citations
Computer Science · #20M10 #20M20 #20M30 #20M35 #Advanced Algebra and Logic #Computability, Logic, AI Algorithms #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2406.18477
openalex publication_date 2024/06/26 · openalex created_date 2024/06/28 · openalex updated_date 2026/08/01
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.