Normal categories of semigroup of order-preserving transformations on a finite chain

K. S. S. Nambooripad intoduced nornal categories to enable to describe the structure of regular semigroups fully. In this paper we describe the ideal categories of the regular semigroup $O{X}_n,$ of non-invertible order-preserving transformations on a finite chain $X_n=\{1\le 2\le \cdots \le n\}$ which are normal categories. Further it is shown that the principal left ideal category of $O{X}_n$ as the power set category $P_o(X_n)$ of $O{X}_n$ and the principal right ideal category as ${\prod }_o(X_n)$ category of ordered partitions of $X_n$ and described the cone semigroup $T\mathcal{L}(O{X}_n)$ and prove that it is isomorphic to $O{X}_n\mathrm{.}$