Quasi-idempotent ranks of some permutation groups and transformation semigroups

Let Sn , An , In , Tn , and Pn be the symmetric group, alternating group, symmetric inverse semigroup, (full) transformations semigroup, and partial transformations semigroup on Xn = {1, . . . , n} , for n ≥ 2 , respectively. A non-idempotent element whose square is an idempotent in Pn is called a quasi-idempotent. In this paper first we show that the quasi-idempotent ranks of Sn (for n ≥ 4) and An (for n ≥ 5) are both 3 . Then, by using the quasi-idempotent rank of Sn , we show that the quasi-idempotent ranks of In , Tn , and Pn (for n ≥ 4) are 4 , 4 , and 5 , respectively.