Southeast Asian Bulletin of Mathematics, 28 (2004), 903-918

In this paper we consider three natural extensions of the inverse monoid POIn of all order preserving
injective partial transformations of a chain with n elements. Namely, the inverse monoids PODIn consisting
of all injective partial transformations on a chain with n elements which are either order preserving or order reversing, the inverse monoid POPIn of all orientation preserving injective partial transformations on a chain with n elements and the inverse monoid PORIn whose elements are all
orientation preserving together with all orientation reversing injective partial transformations on a chain with n elements. Our main aim is to exhibit a presentation for PODIn in terms of n generators and
(n^2 + 7n - 2)/2 relations, a presentation for POPIn in
terms of 2 generators and 2n relations and, finally, a presentation for PORIn in terms of 3 generators and 2n+4 relations.