  Given a locally finite graded set A and a commutative, associative operation
on A that adds degrees, we construct a commutative multiplication * on the set
of noncommutative polynomials in A which we call a quasi-shuffle product; it
can be viewed as a generalization of the shuffle product. The resulting
commutative algebra can be given the structure of a Hopf algebra (_A_,*,Delta).
In the case where A is the set of positive integers and the operation on A is
addition, (_A_,*,Delta) is the Hopf algebra of quasi-symmetric functions. If
rational coefficients are allowed, there is a Hopf algebra isomorphism exp from
the shuffle Hopf algebra on A onto (_A_,*,Delta). We discuss the dual of
(_A_,*,Delta), and define a deformation *_q of * that coincides with * when q =
1 and is isomorphic to the concatenation product when q is not a root of unity.
Finally, we discuss various examples of this construction.
