The aim of this chapter is the description of a method for decomposing an arbitrary transformation semigroup into a wreath product of ‘simpler’ transformation semigroups, namely aperiodic ones and transformation groups. The origin of this theory is the theorem due to Krohn and Rhodes which gave an algorithmic procedure for such a decomposition. There are now various proofs of this result extant, some are set in the theory of transformation semigroups and others are concerned with the theory of state machines. In the light of the close connections between the two theories forged in chapter 2 we can expect a similar correspondence between the two respective decomposition theorems. The proof of the decomposition theorem for state machines has the advantage that it can be motivated the more easily, but at the expense of some elegance. Recently Eilenberg has produced a new, and much more efficient, decomposition and it is this theory that we will now study. It is set in the world of transformation semigroups.

Before we embark on the details let us pause for a moment and consider how we could approach the problem of finding a suitable decomposition. Let M = (Q, Σ,F) be a state machine and let |Q| = n. Consider the collection π of all subsets of Q of order n−1. Then π is an admissible subset system, and we may construct a well-defined quotient machine M/π.