Generalized prelie and permutative algebras

We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of types, we show how to obtain such objects when this set is given an associative commutative product and a second product making it a commutative extended semigroup. Working with a vector space of types, these two products are replaced by a bilinear map $\Phi$ which satisfies a braid equation and a commutation relation. Examples of such structures are defined on sets, semigroups, or groups. These constructions define a family of operads PreLie $\Phi$ which generalize the operad of pre-Lie algebras PreLie. For any embedding from PreLie into PreLie $\phi$ , we construct a family of pairs of cointeracting bialgebras, based on typed and decorated trees: the first coproduct is given by an extraction and contraction process, the types being modified by the action of $\Phi$; the second coproduct is given by admissible cuts, in the Connes and Kreimer's way, with again types modified by the action of $\Phi$. We also study the Koszul dual of PreLie $\Phi$ , which gives generalizations of permutative algebras.