Partial oders on semigroups of partial transformations with restricted range

Let $X$ be any set and $P(X)$ the set of all partial transformations defined on $X$, that is, all functions $\alpha:A\to B$ where $A,B$ are subsets of $X$. Then $P(X)$ is a semigroup under composition. Let $Y$ be a subset of $X$. Recently, Fernandes and Sanwong defined $PT(X,Y)=\{\alpha\in P(X):X\alpha\subseteq Y\}$ and $I(X,Y)$ the set of all injective transformations in $PT(X,Y)$. So $PT(X,Y)$ and $I(X,Y)$ are subsemigroups of $P(X)$. In this paper, we study properties of the so-called natural partial order $\leq$ on $PT(X,Y)$ and $I(X,Y)$ in terms of domains, images and kernels, compare $\leq$ with the subset order, characterize the meet and join of these two orders, then find out elements of $PT(X,Y)$ and $I(X,Y)$ those are compatible. Also, the minimal and maximal elements are described. DOI: 10.1017/S000497271200002