The Krohn-Rhodes Theorem and Local Divisors

TL;DR

This paper presents a new proof of the Krohn-Rhodes Theorem utilizing local divisors, offering a decomposition comparable in size to existing methods, avoiding induction on state set size, and focusing on monoids with group base cases.

Contribution

It introduces a novel proof technique for the Krohn-Rhodes Theorem based on local divisors, simplifying the decomposition process.

Findings

Provides a nearly optimal decomposition size

Avoids induction on state set size

Works exclusively with monoids with group base case

Abstract

We give a new proof of the Krohn-Rhodes Theorem using local divisors. The proof provides nearly as good a decomposition in terms of size as the holonomy decomposition of Eilenberg, avoids induction on the size of the state set, and works exclusively with monoids with the base case of the induction being that of a group.