On isolated subsemigroups of order-decreasing transformation semigroups

For n E N, let Dn be the semigroup of all order-decrasing transformations on Xn = {1, ... , n}, under its natural order. In this paper, we determine isolated, completely isolated, and (left/right) convex subsemigroups of Dn. Furthermore, for {1} =6 U C Xn which contains 1, we find the rank of Dn[U] = {cx E Dn : U C Xn} which is a convex subsemigroup of Dn.