In this paper we calculate presentations for some natural monoids of transformations on
a chain Xn ={1 < 2 < · · · < n}. First we consider ODn [PODn], the monoid of all full
[partial] transformations on Xn that preserve or reverse the order. Two other monoids
of partial transformations on Xn we look at are POPn and PORn—the elements of
the first preserve the orientation and the elements of the second preserve or reverse the
orientation.