  In this paper we introduce the notion of an orientation-preserving
transformation on an arbitrary chain, as a natural extension for infinite
chains of the well known concept for finite chains introduced in 1998 by
McAlister \cite{McAlister:1998} and, independently, in 1999 by Catarino and
Higgins \cite{Catarino&Higgins:1999}. We consider the monoid $\mathscr{POP}(X)$
of all orientation-preserving partial transformations on a finite or infinite
chain $X$ and its submonoids $\mathscr{OP}(X)$ and $\mathscr{POPJ}(X)$ of all
orientation-preserving full transformations and of all orientation-preserving
partial permutations on $X$, respectively. The monoid $\mathscr{PO}(X)$ of all
order-preserving partial transformations on $X$ and its injective counterpart
$\mathscr{POJ}(X)$ are also considered. We study the regularity and give
descriptions of the Green's relations of the monoids $\mathscr{POP}(X)$,
$\mathscr{PO}(X)$, $\mathscr{OP}(X)$, $\mathscr{POPJ}(X)$ and
$\mathscr{POJ}(X)$.
