Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt , Daley  and Deng . It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng .
In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in ,  and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.