Part 7: A new Hopf algebra structure on compositions (quasi-symmetric functions), distinct from the QSym algebra.

Painted trees.

The product.

The coproduct.

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We begin by showing all the painted tree versions of the combinatorial objects of study. This commuting diagram should be compared to Figure 3 in the extended abstract .

The multiplication we define here is analogous to the product in PSym.
In fact, drawing the compositions as "corollas of corollas" with the lower corolla's edges colored, we at first multipy exactly as in PSym.
The rule is that the right operand, which is not split, is painted entirely-- or in effect converted to a painted corolla.

After grafting, the passage to compositions is completed by converting all the painted edges
into a single corolla.

Here is another multiplication, demonstrating that our product is noncommutative.

Here is the same (second above) product, performed on pictures of Boolean subsets. Note that the multiplication order is different. That is because this product is actually being performed in the dual of the algebra of faces of the simplices, as defined in Section 7 of our preprint .

And now the coproduct, performed just as in PSym. The interesting question here, after describing the antipode, is to find the map predicted by the fact that QSym is terminal in the category of coalgebras.

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