1.  There are algebra homomorphisms $\phi_-: H \to \bfk+P(R)$ and $\phi_+: H \to \bfk+(1-P)(R)$ with the decomposition $$\phi=\phi_-^{\star\, (-1)}\star \phi_+,
    \mlabel{eq:phidecom}$$ called the [**Algebraic Birkhoff Decomposition**]{} of $\phi$. Here $\star$ is the convolution product and $\phi_-^{\star\, (-1)}$ is the inverse of $\phi_-$ with respect to $\star$. Further, $$\phi_-(x)=-P\big(\phi(x)+\sum_{(x)} \phi_-(x')\phi(x'')\big)
    \mlabel{eq:phi-}$$ and $$\phi_+(x)=(\id-P)\big(\phi(x)+\sum_{(x)} \phi_-(x')\phi(x'')\big).
    \mlabel{eq:phi+}$$ Here we have used the notation $\Delta(x)=x\ot 1 + 1\ot x + \sum_{(x)} x'\ot x''.$

2.  If $P^2=P$, then the decomposition in Eq. () is unique.

For item (), see [@C-K1] and [@Ma Theorem II.5.1]. For item (), see [@EGK3 Theorem 3.7] where one can also find a proof of item () using Rota-Baxter algebras.

Quasi-shuffle algebras
----------------------

Let $M$ be a commutative semigroup. For each integer $k\geq 0$, let $\bfk M^k$ be the free $\bfk$-module with basis $M^k$, with the convention that $M^0=\{\bfone\}$. Let $$\calh_M=\bigcup_{k=0}^\infty \bfk\, M^k.
 \mlabel{eq:qprod}$$ Following , define the [**quasi-shuffle product**]{} $\msh$ by first taking $\bfone$ to be the multiplication identity. Next for any $m,n\geq 1$ and $\vec a:=(a_1,\cdots , a_m)\in M^{m}$ and $\vec b:=(b_1, \cdots, b_n)\in M^{n}$, denote $\vec a\,'=(a_2,\cdots,a_m)$ and $\vec b\,'=(b_2,\cdots,b_n)$. Recursively define $$\vec{a} \msh \vec{b} =\big(a_1,   \vec{a}\,'\msh \vec b\big )
  + \big(b_1, \vec a \msh \vec{b}\,' \big) \\
  