  We consider the group $(\mathcal{G},*)$ of unitized multiplicative functions
in the incidence algebra of non-crossing partitions, where ``$*$'' denotes the
convolution operation. We introduce a larger group
$(\widetilde{\mathcal{G}},*)$ of unitized functions from the same incidence
algebra, which satisfy a weaker condition of being ``semi-multiplicative''. The
natural action of $\widetilde{\mathcal{G}}$ on sequences of multilinear
functionals of a non-commutative probability space captures the combinatorics
of transitions between moments and some brands of cumulants that are studied in
the non-commutative probability literature. We use the framework of
$\widetilde{\mathcal{G}}$ in order to explain why the multiplication of free
random variables can be very nicely described in terms of Boolean cumulants and
more generally in terms of $t$-Boolean cumulants, a one-parameter interpolation
between free and Boolean cumulants arising from work of Bozejko and
Wysoczanski.
  It is known that the group $\mathcal{G}$ can be naturally identified as the
group of characters of the Hopf algebra Sym of symmetric functions. We show
that $\widetilde{\mathcal{G}}$ can also be identified as group of characters of
a Hopf algebra $\mathcal{T}$, which is an incidence Hopf algebra in the sense
of Schmitt. Moreover, the inclusion of $\mathcal{G}$ into
$\widetilde{\mathcal{G}}$ turns out to be the dual of a natural bialgebra
homomorphism from $\mathcal{T}$ onto Sym.
