Rank Properties of the Semigroup of Singular Transformations on a Finite Set

Abstract

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 Ay\\i k et al. (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.