  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.
