The control units of large digital systems can use up to 80% of the entire hardware implementing the system. Therefore, it is very important to reduce the amount of hardware taken by the control unit and to simplify the design, implementation and verification process. In most cases, the control unit can be constructed as a sequential machine. In this work, a general and unified classification of full-decompositions and formal definitions of different sorts of full-decompositions for Mealy and Moore machines are presented and some theorems about the existence of full-decompositions with the output behaviour realization are formulated and proved. This theorems constitute a theoretical basis for the practical decomposition algorithms and for the software system calculating different sorts of decomposition for sequential machines.