We will introduce a group of semi-multiplicative functions over non-crossing partitions, that has a natural action on sequences of multilinear functionals of a non-commutative probability space. This allows to systemativally study the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature.
The group of semi-multiplicative functions over NC can be seen as a subgroup of an analogue group, where instead of considering functions over the incidence algebra of the poset NC we consider functions over the incidence algebra of all partitions P. This bigger structure allows us to include the classical moment-cumulant formula in the picture. Furthermore, we will identify several other iterative families S, that have sufficient structure to admit a notion of semi-multiplicative functions over S.