The Krohn-Rhodes Theorem and Local Divisors
  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. 
Volker Diekert and Manfred Kufleitner and Benjamin Steinberg
math.GR cs.FL