On the ranks of certain finite semigroups of transformations

TLDR

The finite symmetric group \(S_n\) has rank 2, and the full transformation semigroup on \(n\) symbols has rank 3, generated by the two generators of \(S_n\) together with a defect‑1 element. The semigroup \(\mathrm{Sing}_n\) of singular self‑maps is generated by idempotents, so its idempotent rank is defined as the size of the smallest idempotent generating set. For \(n\ge3\), \(\mathrm{Sing}_n\) has rank \(\tfrac12\,n(n-1)\), and its idempotent rank equals this value.

Abstract

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 (< ∞) symbols has rank 3, being generated by the two generators of 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 <F> = Sing n . This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.