  We define spreadability systems as a generalization of exchangeability
systems in order to unify various notions of independence and cumulants known
in noncommutative probability. In particular, our theory covers monotone
independence and monotone cumulants which do not satisfy exchangeability. To
this end we study generalized zeta and M\"obius functions in the context of the
incidence algebra of the semilattice of ordered set partitions and prove an
appropriate variant of Faa di Bruno's theorem. With the aid of this machinery
we show that our cumulants cover most of the previously known cumulants. Due to
noncommutativity of independence the behaviour of these cumulants with respect
to independent random variables is more complicated than in the exchangeable
case and the appearance of Goldberg coefficients exhibits the role of the
Campbell-Baker-Hausdorff series in this context. In a final section we exhibit
an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants
in a particular spreadability system, thus providing a new derivation of the
Goldberg coefficients.
