Rank properties of the semigroup of singular transformations on a finite set

It is known that the semigroup Sing(n) of all singular self-maps of X-n = {1, 2,...,n} has rank n(n - 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing(n), has the same value as the rank. (See Gomes and Howie, 1987.) Idempotents generating Sing(n) can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Aytk et at (2005). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing(n), defined as the smallest number of (m, r)-path-cycles generating Sing(n), is once again n(n - 1)/2.