On nilpotent subsemigroups of the order-preserving and decreasing transformation semigroups

Let C n be the semigroup of all order-preserving and decreasing transformations on X n = { 1 , … , n } under its natural order, and let N ( C n ) be the subsemigroup of all nilpotent elements of C n . In this paper we determine the minimum generating set of N ( C n ) , and so the rank of N ( C n ) . Moreover, we investigate the nilpotent subsemigroups of C n .