Hierarchy of efficient generators of the symmetric inverse monoid

It is well-known that the symmetric inverse monoid on a set ofn elements can be generated as a semigroup by its group of units and a single element of rankn − 1. We show that the efficiency with which the semigroup is generated in this way depends solely on the index of nilpotence of the rankn − 1 generator. We also investigate the various ways of expressing elements of the semigroup most efficiently as a product of generators.