Tamari's Moebius function and free magmatic algebras

M. Aguiar and F. Sottile proved nice formulas for the coproduct of the Malvenuto-Reutenauer Hopf algebra, resp. for the coproduct of the Hopf algebra of planar binary rooted trees, in terms of the Möbius function of the weak Bruhat order, resp. of the Tamari order.  The free magmatic algebra $Mag(\vert)$ on one element $\vert$ is the vector space spanned by the set of planar binary rooted trees equipped with the grafting of trees. We prove that the Möbius formula for the Tamari order may be described recursively in terms of the grafting.


The space $Mag(\vert)$ has a unique coassociative coalgebra structure $\Delta$ which satisfies the unital infinitesimal relation with respect to the grafting, and the Möbius basis provides an easy way to describe the subspace of primitive elements. On the other hand, the coproduct $\delta $ on planar binary rooted trees satisfies a Joni-Rota infinitesimal relation  with respect to the grafting of trees, so $Mag(\vert)$ has a natural structure of brace algebra.


All these results extend naturally to the vector space $\mathbb{K}[\text{PBT}^S]$ generated by the set of planar binary rooted trees with the internal nodes colored by the elements of some set $S$, that is the free algebra generated by $\vert S\vert$ magmatic products. So, using the natural action of $\mathbb{K}[\text{PBT}^S]$ on any $S$-magmatic algebra, we get a structure theorem for this type of algebras, whenever they are equipped with a coalgebra structure which satisfies the unital infinitesimal relation with respect to all the magmatic products.


Finally we show examples of this type of generalized bialgebras. Note that our theory applies even if some of the products are associative, it suffices that two products $\cdot_1$ and  
$\cdot_2$ satisfy that
$$\cdot_1\circ (\cdot_2\otimes \text{{Id}}) \neq \cdot_2\circ (\text{{Id}}\otimes \cdot_1),$$
to get non trivial Möbius operations on their subspaces of primitive elements.