Let \(\cal{O}_{n}\) be the semigroup of all order-preserving (full) transformations on the finite chain Xn = {1,…,n} under its natural order. For a singular idempotent ξ, it is shown that \({\cal{O}_{n}(\xi)=\{\alpha \in \cal{O}_{n}:\alpha^{m}=\xi} \ {\text{for some}}\ m \in \mathbb{N}\}\) is a maximal nilpotent subsemigroup of \(\cal{O}_{n}\) with zero ξ. Moreover, for a nonempty subset Y of Xn, we give a necessary and sufficient condition for the set \({\cal{O}_{n}}(Y)\) to be a subsemigroup. Then we find a unique minimal generating set, and so rank, of \({\cal{O}_{n}}(Y)\) whenever it is a subsemigroup of \({\cal{O}_{n}}\) . Every subset Y of Xn such that \({\cal{O}_{n}}(Y)\) is (completely) isolated was characterized.
