On the rank of transformation semigroup T(n,m)

Abstract

Let Tn and Sn be the full transformation semigroup and the symmetric group on Xn = {1, . . . , n},respectively. For $n, m ∈ Z+ with m ≤ n − 1 let T(n,m) = {α ∈ Tn : Xmα = Xm}.In this paper we research generating sets and the rank of T(n,m) . In particular, we prove thatrank $(T(n,m)) =   2 if (n, m) = (2, 1) or (3, 2) 3 if (n, m) = (3, 1) or 4 ≤ n and m = n − 14 if 4 ≤ n and 1 ≤ m ≤ n − 2$.for 1 ≤ m ≤ n − 1.