On factorisations and generators in transformation semigroups

It is shown that the classical decomposition of permutations into disjoint cycles can be extended to more general mappings by means of path-cycles, and an algorithm is given to obtain the decomposition. The device is used to obtain information about generating sets for the semigroup of all singular selfmaps of X-n = {1, 2,..., n}. Let T-n,T-r = S-n boolean OR K-n,K-r, where S-n is the symmetric group and K-n,K-r is the set of maps alpha : X-n --> X-n such that \im((alpha)\ <= r. The smallest number of elements of K-n,K-r which, together with S-n, generate T-n,T-r is p(r)(n), the number of partitions of n with r terms.