The Krohn-Rhodes theorem and local divisors (Q2893300)

Given a monoid \(M\) and its arbitrary element \(c \in M\), the monoid defined on the set \(cM \cap Mc\) by the multiplication rule \(cm \circ cn = cmn\) is called a local divisor of \(M\). The main result of this paper asserts that every transformation monoid \(M\) generated by a set \(A\) divides a wreath product of the local divisor of \(M\) corresponding to a given generator \(c \in A\) and the submonoid of \(M\) generated by \(A \setminus \{c\}\) (with the help of additional constant transformations). This provides a new, short proof of the Krohn-Rhodes theorem.