On the rank and (0, 1)- square idempotent rank of the semigroup G(n, r)

We introduce one-to-one singular transformation semigroup of ascending-preserving finite parts, By analyzing the \left( 0,1 \right)- square idempotent elements and star Green's relations, the unique minimal \left( 0,1 \right)- square Idempotent generating set, rank and \left( 0,1 \right)- square Idempotent rank of the semigroup G\left( n,r \right) areobtained, respectively. Furthermore, the relative rank of the semigroup G\left( n,r \right) with respect to itself each star ideal G\left( n,l \right) is determined if 0 \leqslant l \leqslant r.