  We study three different poset structures on the set of all compositions. In
the first case, the covering relation consists of inserting a part of size one
to the left or to the right, or increasing the size of some part by one. The
resulting poset was studied by the author in "A poset classifying
non-commutative term orders", and then in "Standard paths in another
composition poset" where some results about generating functions for standard
paths in this poset was established.
  The latter article was inspired by the work of Bergeron, Bousquet-M{\'e}lou
and Dulucq on "Standard paths in the composition poset", where they studied a
poset where there are additional cover relations which allows the insertion of
a part of size one anywhere in the composition. Finally, following a suggestion
by Richard Stanley we study yet a third which is an extension of the previous
two posets. This poset is related to quasi-symmetric functions.
  For these posets, we study generating functions for saturated chains of fixed
width k. We also construct ``labeled'' non-commutative generating functions and
their associated languages.
