On the ranks of certain finite semigroups of transformations

It is well-known (see [2]) that the finite symmetric group S n has rank 2. Specifically, it is known that the cyclic permutations generate S n ,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n ( S n , together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Sing n , the semigroup of all singular self-maps of {1, …, n }, is harder to determine: in Section 2 it is shown to be ½ n(n − 1) (for n ≽ 3). The semigroup Sing n it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Sing n as the cardinality of the smallest possible set P of idempotents for which = Sing n . This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.