the submonoid of $\T(X)$ of all orientation-preserving full transformations on $X$. An equivalent notion of an orientation-preserving transformation was first introduced by McAlister in [@McA] and, independently, by Catarino and Higgins in [@CH]. It is clear that $\O(X)$ is a submonoid of $\OP(X)$, i.e. $\O(X)\subset \OP(X)\subset \T(X)$. It is interesting to note that the relative rank of $\T(X)$ modulo $\OP(X)$ as well as the relative rank of $\OP(X)$ modulo $\O(X)$ is one (see [@CH; @HRH]), but the situation will change if one considers transformations with restricted range.

Let $Y=\{a_{1}<a_{2}<\cdots <a_{m}\}$ be a nonempty subset of $X$, for a natural number $m\leq n$, and denote by $\T(X,Y)$ the subsemigroup $\{\alpha \in \T(X) : X \alpha \subseteq Y\}$ of $\T(X)$ of all transformations with range (image) restricted to $Y$. The set $\T(X,Y)$ coincides with $\T(X)$, whenever $Y=X$ (i.e. $m=n$). In 1975, Symons [@Symons] introduced and studied the semigroup $\T(X,Y)$, which is called semigroup of transformations with restricted range. Recently, the rank of $\T(X,Y)$ was computed by Fernandes and Sanwong in [@FS]. They proved that the rank of $\T(X,Y)$ is the Sterling number $S(n,m)$ of second kind with $\left\vert X\right\vert =n$ and $\left\vert Y\right\vert =m$. The rank of the order-preserving counterpart $\O(X,Y)$ of 