  Let A and B be two connected graded commutative k-algebras of finite type,
where k is a perfect field of positive characteristic p. We prove that the
quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if
and only if A and B are isomorphic as graded k-vector spaces equipped with a
Frobenius (pth-power) map.
  For the hardest part of this analysis, we work with the dual construction,
and are led to study connected graded cocommutative Hopf algebras H with two
additional properties: H is free as an associative algebra, and the projection
onto the indecomposables is split as a morphism of graded k-vector spaces
equipped with a Verschiebung map.
  Building on work on non-commutative Witt vectors by Goerss, Lannes, and
Morel, we classify such free, `split' Hopf algebras.
  A topological consequence is that, if X is a based path connected space, then
the Hopf algebra H^*(Omega Sigma X;k) is determined by the stable homotopy type
of X.
  We also discuss the much easier analogous characteristic 0 results, and give
a characterization of when our quasi--shuffle algebras are polynomial,
generalizing the so-called Ditters conjecture.
