Generalized “IMPLICATION” operation (IO) is one of the logical operations that widely exist in experienced thinking, uncertain reasoning, and all kinds of multi-valued logical systems and have general significance. But the applications often give the logical operators without theoretic guide and the analyses of their effectiveness. In addition, they are often given at will and blindly. The authors first study the thinking foundation of IO, hold that IO is the inverse operation of series reasoning operation, then put forward the IO axiom, give the definition of IO from the viewpoint of algebraic system, raise and prove the representation theorem of IO which guarantees that the operators generated by it belong to IO and all operators belonging to IO can be generated by it, compare and analyse the implication operators in common use, finally study the utilization of IO in series reasoning operation. Thus the faults that the existing theory about IO have been overcome, the applications can design the implication operators according to the IO axiom and the representation theorem of IO which provide the theoretic foundation for the designing of generalized implication operators and ensure the reasoning conclusions exact and believable.