Naturally ordered transformation semigroups

The full transformation semigroup (T _x o) on the setX is studied with respect to the natural partial order which is defined on any regular semigruop. A characterization of this ordering in terms of images and kernels is given, the maximal and minimal elements are determined and the covering elements are described. Lower and upper bounds for two transformations are studied and necessary as well as sufficient conditions for their existence are gieven.