On the semigroup of all injective orientation-preserving and order-decreasing partial transformations on a finite chain

Let POPDZn ( PODZ n ) be the semigroup consisting of all injective orientation-preserving (order-preserving) and order-decreasing transformations on the finite chain { 1 < < n } . For 1 <= r <= n , let PODZ(n,r) = {alpha is an element of PODZn : | im (alpha)| <= r } and POPDZ(n,r) = {alpha is an element of POPDZn : | im (alpha)| <= r } . In this paper, we find the minimal generating sets, and so the ranks of PODZ(n, r) and POPDZ(n, r) . We also consider the nilpotent subsemigroup of PODZ(n, r) .