  In the last decennia two generalizations of the Hopf algebra of symmetric
functions have appeared and shown themselves important, the Hopf algebra of
noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric
functions QSymm. It has also become clear that it is important to understand
the noncommutative versions of such important structures as Symm the Hopf
algebra of symmetric functions. Not least because the right noncommmutative
versions are often more beautiful than the commutaive ones (not all cluttered
up with counting coefficients). NSymm and QSymm are not truly the full
noncommutative generalizations. One is maximally noncommutative but
cocommutative, the other is maximally non cocommutative but commutative. There
is a common, selfdual generalization, the Hopf algebra of permutations of
Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best understood
as a Hopf algebra of endomorphisms. In any case, this point of view suggests
vast generalizations leading to the Hopf algebras of endomorphisms and word
Hopf algebras with which this paper is concerned. This point of view also sheds
light on the somewhat mysterious formulas of MPR and on the question where all
the extra structure (such as autoduality) comes from. The paper concludes with
a few sections on the structure of MPR and the question of algebra retractions
of the natural inclusion of Hopf algebras of NSymm into MPR and section of the
naural projection of MPR onto QSymm.

Quantum Algebra;Combinatorics