  Two important generalizations of the Hopf algebra of symmetric functions are
the Hopf algebra of noncommutative symmetric functions and its graded dual the
Hopf algebra of quasisymmetric functions. A common generalization of the latter
is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter
Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra.
That turns out to be a fruitful way of looking at things and gives rise to wide
ranging further generalizations such as the word Hopf algebra and the double
word Hopf algebra (and many more).
