  Observables of quantum or classical mechanics form algebras called quantum or
classical Hamilton algebras respectively (Grgin E and Petersen A (1974) {\it J
Math Phys} {\bf 15} 764\cite{grginpetersen}, Sahoo D (1977) {\it Pramana} {\bf
8} 545\cite{sahoo}). We show that the tensor-product of two quantum Hamilton
algebras, each characterized by a different Planck's constant is an algebra of
the same type characterized by yet another Planck's constant. The algebraic
structure of mixed quantum and classical systems is then analyzed by taking the
limit of vanishing Planck's constant in one of the component algebras. This
approach provides new insight into failures of various formalisms dealing with
mixed quantum-classical systems. It shows that in the interacting mixed
quantum-classical description, there can be no back-reaction of the quantum
system on the classical. A natural algebraic requirement involving restriction
of the tensor product of two quantum Hamilton algebras to their components
proves that Planck's constant is unique.
