Terseness: Minimal idempotent generating sets for K(n,r)

Let n and r be positive integers with 1 < r < n and let K(n,r) consist of all transformations on X _n = {1,...,n} having image size less than or equal to r. For 1 < r < n, there exist rank-r elements of K(n,r) which are not the product of two rank-r idempotents. With this limitation in mind, we prove that for fixed r, and for all n large enough relative to r, that there exists a minimal idempotent generating set U of K(n,r) such that all rank-r elements of K(n,r) are contained in U ³. Moreover, for all n > r > 1, there exists a minimal idempotent generating set W for K(n,r) such that not every rank-r element is contained in W ³.