On the ranks of certain finite semigroups of transformations
Gracinda M. S. Gomes, John M Howie · Mathematical Proceedings of the Cambridge Philosophical Society · 1987
It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.