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The product of functions is commutative - fg and gf are the same thing. But one can now consider set of functions that have all the required extra structure to make them set of functions acting over a group - except for the fact that the product is now not commutative anymore. Then the elements cannot be functions on a group anymore -- in fact they can't be functions at all. But one can still pretend that they are “functions” on some funny type of set: A quantum group.