Bilateral semidirect product decompositions of transformation monoids

In this paper we consider the monoid ORn of all full transformations on a chain with n elements that preserve or reverse the orientation, as well as its submonoids ODn of all order-preserving or order-reversing elements, OPn of all orientation-preserving elements and O-n of all order-preserving elements. By making use of some well known presentations, we show that each of these four monoids is a quotient of a bilateral semidirect product of two of its remarkable submonoids.