  For any set S, the free magmatic algebra spanned by card(S) binary products
is the vector space spanned by the set of all planar rooted binary trees with
the internal nodes colored by the elements of S, graded by the number of leaves
of a tree. We show that it has a unique structure of coassociative coalgebra
such that the coproduct satisfies the unital infinitesimal condition with each
magmatic product, and prove an analog of Aguiar-Sottile formula in this
context, describing the coproduct in terms of the Moebius basis for the Tamari
order. The last result allows us to compute the subspace of primitive elements
of any unital infinitesimal S-magmatic bialgebra. As an example, we construct a
set of generators of the dual of Pilaud and Pons bialgebra of integer relations
and compute an explicit basis of its subspace of primitive elements.
