Relative rank of transformation semigroups

Let P_n, T_n, S_n and A_n be the partial transformations semigroup, the full transformations semigroup, the symmetric group and the alternating group on a set X_n={1,…,n}, respectively. For 1≤r≤n, let K_(n,r) &=&{α∈T_n: "im" (α)≤r}, T_(n,r)&=&S_n∪K_(n,r), A_(n,r)&=&A_n∪K_(n,r), PK_(n,r)&=&{α∈P_n: "im" (α)≤r}, PT_(n,r)&=&S_n∪PK_(n,r), PA_(n,r)&=&A_n∪PK_(n,r).) For 1≤r≤n-1, it is found that the necessary and sufficient conditions for any subset of PK_(n,r) (K_(n,r)) to be a (minimal) relative generating set of PT_(n,r) (T_(n,r)) modulo S_n and it is given that the relative rank of PT_(n,r) and T_(n,r) modulo S_n. Also, it is examined that the similar results for the semigroups PA_(n,r) and A_(n,r) by taking A_n instead of S_n.