the present paper, we will determine the relative rank for a particular semigroup of transformations on a finite set.

Let $X$ be a finite chain, say $X=\{1<2<\cdots <n\}$, for a natural number $n$. A transformation $\alpha \in \T(X)$ is called *order-preserving* if $x\le y$ implies $x\alpha\le y\alpha$, for all $x,y\in X$. We denote by $\O(X)$ the submonoid of $\T(X)$ of all order-preserving full transformations on $X$. The relative rank of $\T(X)$ modulo $\O(X)$ was considered by Higgins, Mitchell, and Ruškuc in [@HMR]. They showed that $\rank(\T(X) : \O(X)) = 1$, when $X$ is an arbitrary countable chain or an arbitrary well-ordered set, while $\rank(\T(\R) : \O(\R))$ is uncountable, by considering the usual order of the set $\R$ of real numbers. In [@DFK], Dimitrova, Fernandes, and Koppitz studied the relative rank of the semigroup $\O(X)$ modulo $J=\{\alpha\in\O(X) : |X\alpha|=|X|\}$, for an infinite countable chain $X$. We say that a transformation $\alpha \in\T(X)$ is *orientation-preserving* if there are subsets $X_{1},X_{2}\subseteq X$ with $\emptyset \neq X_{1}<X_{2}$, (i.e. $x_{1}<x_{2}$ for $x_{1}\in X_{1}$ and $x_{2}\in X_{2}$), $X=X_{1}\cup X_{2}$, and $x\alpha \leq y\alpha$, whenever either $(x,y)\in X_{1}^{2}\cup X_{2}^{2}$ with $x\leq y$ or $(x,y)\in X_{2}\times X_{1}$. Note that $X_{2}=\emptyset$ provides $\alpha \in \O(X)$. We denote by $\OP(X)$ 