We show how an equivalence relation on the set of firing sequences of a general net can be defined so that equivalent sequences have the same set of processes. For a special class of nets, we establish a characterization of equivalence classes on firing sequences which define the same partial orders on transition occurrences as those defined by the processes. With every n-safe net we associate a 1-safe net which has, in a certain sense, the same set of processes as the initial net. For some classes of nets we obtain some kind of regular properties for the set of processes.