Order-decreasing transformation semigrups

Let X_n denote the chain \left\{1,2,\ldots,n\right\} under its natural order, and let P_n be the semigroup consisting of all partial transformations on X_n. The subsemigroup of P_n consisting of all (full) transformations is denoted by T_n, the subsemigroup of P_n consisting of all order-decreasing partial transformations is denoted by {PD}_n, and the subsemigroup of P_n consisting of all order-decreasing transformations is denoted by D_n. In this study, we examine some important subsets of order-decreasing transformation semigroups, especially idempotent and nilpotent sets, as well as their generating sets and ranks, focusing on some significant algebraic properties.