Some remarks of finite full transformation semigroups (Q1098942)

Let \(X=\{1,2,...,n\}\) and denote by \(S_ n\) the semigroup of all singular transformations of X. J. M. Howie showed a long time ago that \(S_ n\) is generated by its subset E of idempotents so that for each element \(\alpha \in S_ n\) there exists a unique positive integer \(4k(\alpha)\) such that \(\alpha \in E^{k(\alpha)}\) but \(\alpha \not\in E^{k(\alpha)-1}\). More recently, Howie showed that, in fact, \(k(\alpha)\leq \) rank\(\alpha +orb \alpha -fix \alpha\) where rank\(\alpha = | X\alpha |\), fix\(\alpha = |\{x\in X:x\alpha = x\}|\) and orb\(\alpha\) is the number of nonsingleton orbits of \(\alpha\). The author associates with each \(\alpha \in S_ n\) collections \(T_ k\) of mutually disjoint subsets of X. For each of these he produces a positive integer (there are a number of preliminary steps involved in getting it) and then he takes \(m(\alpha)\) to be the minimum of all these numbers. The main theorem expresses the fact that \(\alpha\) can be written as the product of \(m(\alpha)\) idempotents.