On orientation-preserving transformations of a chain

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 and, independently, in 1999 by Catarino and Higgins. We consider the monoid POP(X)(X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids and POPJ(X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively. The monoid OPPO(X) of all order-preserving partial transformations on X and its injective counterpart are also considered. We study the regularity and give descriptions of the Green's relations of the monoids POP(X), PO(X), OP(X), POPJ(X), and POJ(X).